7e maths for maths for technicians - mcgraw...

MATHS TECHNICIANS

FOR

7e

BLAIR ALLDIS VINCENT KELLY

MATHS TECHNICIANSFOR

7e

The seventh edition of Mathematics for Technicians retains all the popular features and continues the successful building block approach of the previous editions which has made this the leading Australian text for students completing the Engineering Training Package.

New to the seventh edition and in keeping with the author’s philosophy, that teaching maths can be an enjoyable experience for the student, you will find:

n additional reality based scenarios that increase the relevance of mathematics to the student’s future careers.

n a new section on Statistics in line with the recent changes to the Engineering Training Package

n updated examples and exercises to all chapters

n cross references to additional exercises, extension material and testbanks on the Online Learning Centre

Get the most from your course. The online Learning Centre provides additional resources and a powerful learning experience. The OLC for students and instructors is available at www.mhhe.com/au/MFT7e

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Copyright © 2012 McGraw-Hill Australia Pty Limited.

Every effort has been made to trace and acknowledge copyrighted material. The authors and publishers tender their apologies should any infringement have occurred.

Reproduction and communication for educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this work, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the institution (or the body that administers it) has sent a Statutory Educational notice to Copyright Agency Limited (CAL) and been granted a licence. For details of statutory educational and other copyright licences contact: Copyright Agency Limited, Level 15, 233 Castlereagh Street, Sydney NSW 2000. Telephone: (02) 9394 7600. Website: www.copyright.com.au

Reproduction and communication for other purposes Apart from any fair dealing for the purposes of study, research, criticism or review, as permitted under the Act, no part of this publication may be reproduced, distributed or transmitted in any form or by any means, or stored in a database or retrieval system, without the written permission of McGraw-Hill Australia including, but not limited to, any network or other electronic storage.

Enquiries should be made to the publisher via www.mcgraw-hill.com.au or marked for the attention of the permissions editor at the address below.

National Library of Australia Cataloguing-in-Publication Data Author: Alldis, Blair K. (Blair Knox) Title: Mathematics for technicians / Blair Alldis, Vincent Kelly.

Edition: 7th ed. ISBN: 9781743070772 (pbk.) Notes: Includes index. Subjects: Shop mathematics—Problems, exercises, etc. Engineering mathematics—Problems, exercises, etc. Other Authors/Contributors:

Kelly, Vincent Ambrose. Dewey Number: 510.2462

Published in Australia by McGraw-Hill Australia Pty Ltd Level 2, 82 Waterloo Road, North Ryde NSW 2113 Publisher: Norma Angeloni Tomaras Senior production editor: Yani Silvana Copyeditor: Ross Blackwood Proofreader: Pauline O’Carolan Indexer: Shelley Barons Cover and internal design: George Creative Typeset in Berkeley Book 10.5/14.5pt by Laserwords Private Limited, India Printed in on by e.g. Printed in China on 70gsm matt art by iBook Printing Ltd

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Enquiries should be made to the publisher via www.mcgraw-hill.com.au or marked for the attention of the permissions editor at the

Sample

Enquiries should be made to the publisher via www.mcgraw-hill.com.au or marked for the attention of the permissions editor at the

Sample

National Library of Australia Cataloguing-in-Publication Data

Sample

National Library of Australia Cataloguing-in-Publication Data

Title: Mathematics for technicians / Blair Alldis, Vincent Kelly.

Sample

Title: Mathematics for technicians / Blair Alldis, Vincent Kelly.

Subjects: Shop mathematics—Problems, exercises, etc.

Sample

Subjects: Shop mathematics—Problems, exercises, etc. Engineering mathematics—Problems, exercises, etc.

Sample

Engineering mathematics—Problems, exercises, etc. Other Authors/Contributors:

Sample

Other Authors/Contributors:

Kelly, Vincent Ambrose.

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Kelly, Vincent Ambrose.

Sample

Dewey Number: 510.2462 Sample

Dewey Number: 510.2462 Sample

Published in Australia by Sample

Published in Australia by McGraw-Hill Australia Pty Ltd Sam

ple

McGraw-Hill Australia Pty Ltd Level 2, 82 Waterloo Road, North Ryde NSW 2113

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Level 2, 82 Waterloo Road, North Ryde NSW 2113

only

(the Act) allows a maximum of one chapter or 10% of the pages of this work, whichever is

only

(the Act) allows a maximum of one chapter or 10% of the pages of this work, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the

only

the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the institution (or the body that administers it) has sent a Statutory Educational notice to Copyright Agency Limited (CAL) and been

only

institution (or the body that administers it) has sent a Statutory Educational notice to Copyright Agency Limited (CAL) and been granted a licence. For details of statutory educational and other copyright licences contact: Copyright Agency Limited, Level 15, 233

onlygranted a licence. For details of statutory educational and other copyright licences contact: Copyright Agency Limited, Level 15, 233

Castlereagh Street, Sydney NSW 2000. Telephone: (02) 9394 7600. Website: www.copyright.com.au

onlyCastlereagh Street, Sydney NSW 2000. Telephone: (02) 9394 7600. Website: www.copyright.com.au

Apart from any fair dealing for the purposes of study, research, criticism or review, as permitted under the Act, no part of this

onlyApart from any fair dealing for the purposes of study, research, criticism or review, as permitted under the Act, no part of this

publication may be reproduced, distributed or transmitted in any form or by any means, or stored in a database or retrieval system, only

publication may be reproduced, distributed or transmitted in any form or by any means, or stored in a database or retrieval system, without the written permission of McGraw-Hill Australia including, but not limited to, any network or other electronic storage. on

lywithout the written permission of McGraw-Hill Australia including, but not limited to, any network or other electronic storage.

Enquiries should be made to the publisher via www.mcgraw-hill.com.au or marked for the attention of the permissions editor at the only

Enquiries should be made to the publisher via www.mcgraw-hill.com.au or marked for the attention of the permissions editor at the only

Contents V

CONTENTS

1 FRACTIONS AND DECIMALS 21.1 Integers 21.2 Order of operations 31.3 Fractions 31.4 Decimal fractions 71.5 Rounding 101.6 Significant figures 111.7 Using a scientific calculator 141.8 Evaluating mathematical functions using a

calculator 15 Self-test 15

2 RATIO, PROPORTION AND PERCENTAGE 17

2.1 Ratio 172.2 Direct variation: the unitary method 192.3 Direct variation: the algebraic method 212.4 Inverse variation 242.5 Joint variation 262.6 Percentages 27 Self-test 29

3 MEASUREMENT AND MENSURATION 303.1 SI units (système internationale d’unités) 303.2 Estimations 323.3 Approximations 333.4 Accuracy of measurement 333.5 Stated accuracy 353.6 Implied accuracy in a stated measurement 353.7 Additional calculator exercises involving

squares and square roots 363.8 Pythagoras’ theorem 373.9 ‘Ideal’ figures 383.10 Perimeters and areas of right-angled triangles,

quadrilaterals and polygons 403.11 The area of a rectangle 40

6 INTRODUCTION TO GEOMETRY 946.1 Points, lines, rays and angles 946.2 Angles 956.3 Complements and supplements 966.4 Vertically opposite angles 986.5 Parallel lines and a transversal 98

3.12 The area of a triangle 413.13 The circle: circumference and area 443.14 Volumes of prisms 453.15 Surface areas of prisms 47 Self-test 49

4 INTRODUCTION TO ALGEBRA 504.1 Substitutions 504.2 Addition of like terms 524.3 Removal of brackets 534.4 Multiplication and division of terms 534.5 The distributive law 544.6 HCF and LCM 564.7 Algebraic fractions 57

4.8 Algebraic fractions of the form a 6 b

c 594.9 Two important points 604.10 Solving linear equations 624.11 Solving more difficult linear equations 654.12 Using linear equations to solve practical

problems 664.13 Simultaneous linear equations 694.14 The substitution method 704.15 The elimination method 714.16 Practical problems 73 Self-test 75

5 FORMULAE: EVALUATION AND TRANSPOSITION 77

5.1 Evaluation of the subject of a formula 775.2 More SI units 795.3 Evaluation of a formula 805.4 Transposition 835.5 Transpositions in which grouping is required 865.6 More transposition and evaluation 88 Self-test 91

6.6 The angles of a triangle 996.7 The use of the rule, protractor and set

square 1006.8 Geometrical constructions using a

compass 103 Self-test 105

PART 1 NUMERACY AND ALGEBRA

PART 2 GEOMETRY AND TRIGONOMETRY

About the authors viiiAcknowledgments viiiPreface ixText at a glance xE-student/E-instructor xiii

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4.5 The distributive law 54

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4.5 The distributive law 54

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3 MEASUREMENT AND MENSURATION 30

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3 MEASUREMENT AND MENSURATION 303.1 SI units (système internationale d’unités) 30

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3.1 SI units (système internationale d’unités) 30

3.4 Accuracy of measurement 33

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3.4 Accuracy of measurement 333.5 Stated accuracy 35 Sam

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3.5 Stated accuracy 353.6 Implied accuracy in a stated measurement 35Sam

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3.6 Implied accuracy in a stated measurement 35Additional calculator exercises involving Sam

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Additional calculator exercises involving squares and square roots 36Sam

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squares and square roots 36

4.6 HCF and LCM 56

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4.6 HCF and LCM 564.7 Algebraic fractions 57

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4.7 Algebraic fractions 57

4.8 Algebraic fractions of the form

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4.8 Algebraic fractions of the form 4.9 Two important points 60

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4.9 Two important points 604.10 Solving linear equations 62

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4.10 Solving linear equations 624.11 Solving more difficult linear equations 65

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4.11 Solving more difficult linear equations 654.12 Using linear equations to solve practical

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4.12 Using linear equations to solve practical

4.13 Simultaneous linear equations 69

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4.13 Simultaneous linear equations 69

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3.13 The circle: circumference and area 44

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3.13 The circle: circumference and area 443.14 Volumes of prisms 45

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3.14 Volumes of prisms 453.15 Surface areas of prisms 47

only3.15 Surface areas of prisms 47

4 INTRODUCTION TO ALGEBRA 50

only4 INTRODUCTION TO ALGEBRA 50

4.1 Substitutions 50

only4.1 Substitutions 50

4.2 Addition of like terms 52

only4.2 Addition of like terms 52

4.3 Removal of brackets 53only

4.3 Removal of brackets 534.4 Multiplication and division of terms 53on

ly4.4 Multiplication and division of terms 534.5 The distributive law 54on

ly4.5 The distributive law 54on

ly4.6 HCF and LCM 56on

ly4.6 HCF and LCM 564.7 Algebraic fractions 57on

ly4.7 Algebraic fractions 57

ContentsVI

7 GEOMETRY OF TRIANGLES AND QUADRILATERALS 107

7.1 Classification of triangles 1077.2 Detecting equal angles 1097.3 The incentre, the circumcentre, the centroid

and the orthocentre 1107.4 Congruent triangles 1127.5 Similar triangles 1147.6 Special quadrilaterals and their properties 1177.7 Hero’s formula 1207.8 Areas of the parallelogram and the rhombus 121 Self-test 124

8 GEOMETRY OF THE CIRCLE 1268.1 The circle: some terms used 1268.2 The circle: proportional relationships 1288.3 Angles associated with the circle 1298.4 Chords of a circle 1338.5 Tangents to circles 1348.6 The angle in the alternate segment 1368.7 Intersecting circles 1378.8 Practical situations 138 Self-test 139

9 STRAIGHT LINE COORDINATE GEOMETRY 142

9.1 The number plane 1429.2 The gradient of a line 1449.3 Intercepts 147

PART 3 APPLIED MATHEMATICS

9.4 The straight line equation y 5 mx 1 b 1489.5 Finding the equation for a particular straight

line 1519.6 Dependent and independent variables 1529.7 Plotting and interpolation 1549.8 Plotting graphs from equations 1589.9 Graphical solution of linear equations and

systems of two (simultaneous) linear equations 160

Self-test 163

10 INTRODUCTION TO TRIGONOMETRY 16610.1 Conversion of angles between sexagesimal

measure (degrees, minutes and seconds) and decimal degrees 166

10.2 The tangent ratio: construction and definition 167

10.3 The tangent ratio: finding the length of a side of a right-angled triangle 169

10.4 The tangent ratio: evaluating an angle 17010.5 The sine and cosine ratios 17110.6 Applications 17410.7 How to find a trigonometrical ratio from one

already given 17710.8 Compass directions; angles of elevation and

depression 17810.9 Area of a triangle 180 Self-test 181

11 INDICES AND RADICALS 18411.1 Radicals 18411.2 Positive integral indices 18611.3 Substitutions and simplifications 18711.4 The four laws for indices 18811.5 Zero, negative and fractional indices 19111.6 Scientific notation 19411.7 Engineering notation 19611.8 Transposition in formulae involving exponents

and radicals 198 Self-test 199

12 POLYNOMIALS 20112.1 Definition 20112.2 Multiplication of polynomials 20212.3 Factorising 20312.4 Common factors 20412.5 The difference of two squares 20512.6 Trinomials 20612.7 Composite types 21012.8 Algebraic fractions (simplification by

factorising) 21012.9 Quadratic equations: introduction 21112.10 Solution of the equation p 3 q 5 0 21212.11 Solution of the equation x2 5 c 21212.12 Solution of the equation ax2 1 bx 5 0 21412.13 Solution of the equation ax2 1 bx 1 c 5 0 21512.14 ‘Roots’ and ‘zeros’ 21712.15 The quadratic formula 217

12.16 Graphical solution of a quadratic equation 223 Self-test 224

13 FUNCTIONS AND THEIR GRAPHS 22513.1 Function notation 22513.2 The parabola: graphing the curve from its

equation 22713.3 The parabola: finding the equation knowing

the coordinates of three points 23113.4 Practical applications of the parabola 23213.5 The circle 23413.6 The rectangular hyperbola 23613.7 Algebraic solution of simultaneous equations

that involve quadratic functions 23713.8 Graphical solution of simultaneous equations

that involve quadratic functions 23813.9 Verbally formulated problems involving

finding the maximum or minimum value of a quadratic function by graphical means 240

Self-test 242

14 LOGARITHMS AND EXPONENTIAL EQUATIONS 244

14.1 Definition of a logarithm: translation between exponential and logarithmic languages 244

14.2 Evaluations using the definition (logs and antilogs) 245

14.3 Powers and logarithms with base 10 or e 24614.4 Decibels 247

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10.9 Area of a triangle 180

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10.9 Area of a triangle 180 Self-test 181

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Self-test 181

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11 INDICES AND RADICALS 184

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11 INDICES AND RADICALS 184

11.2 Positive integral indices 186

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11.2 Positive integral indices 18611.3 Substitutions and simplifications 187

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11.3 Substitutions and simplifications 18711.4 The four laws for indices 188

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11.4 The four laws for indices 18811.5 Zero, negative and fractional indices 191

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11.5 Zero, negative and fractional indices 19111.6 Scientific notation 194

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11.6 Scientific notation 19411.7 Engineering notation 196Sam

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11.7 Engineering notation 19611.8 Transposition in formulae involving exponents Sam

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11.8 Transposition in formulae involving exponents Sample

and radicals 198Sample

and radicals 198 Self-test 199 Sam

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Self-test 199

12 POLYNOMIALS 201Sample

12 POLYNOMIALS 201

12.16 Graphical solution of a quadratic equation 223

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12.16 Graphical solution of a quadratic equation 223

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10.2 The tangent ratio: construction and

only

10.2 The tangent ratio: construction and definition 167

only

definition 167The tangent ratio: finding the length of a side

only

The tangent ratio: finding the length of a side of a right-angled triangle 169

onlyof a right-angled triangle 169

10.4 The tangent ratio: evaluating an angle 170

only10.4 The tangent ratio: evaluating an angle 170

10.5 The sine and cosine ratios 171

only10.5 The sine and cosine ratios 171

10.6 Applications 174

only10.6 Applications 174

10.7

only10.7 How to find a trigonometrical ratio from one

onlyHow to find a trigonometrical ratio from one

already given 177only

already given 17710.8 Compass directions; angles of elevation and on

ly10.8 Compass directions; angles of elevation and

depression 178only

depression 17810.9 Area of a triangle 180on

ly10.9 Area of a triangle 180 Self-test 181on

ly Self-test 181

Contents VII

14.5 The three laws of logarithms 25014.6 Change of base 25314.7 Evaluations using the laws of logarithms 25414.8 Solution of logarithmic equations 25514.9 Exponential equations 25514.10 Change of subject involving logarithms 25814.11 Applications 25814.12 The curve y 5 k logb Cx 25914.13 A quick but important revision of exponents

and logarithms 26314.14 Limits 264 Self-test 265

15 NON-LINEAR EMPIRICAL EQUATIONS 26715.1 Introduction 26715.2 Conversion to linear form by algebraic

methods 267 15.3 The use of logarithms to produce a linear

form 269 Self-test 272

16 COMPOUND INTEREST: EXPONENTIAL GROWTH AND DECAY 273

16.1 Compound interest 27316.2 Exponential growth 27416.3 Graphs of exponential functions 27816.4 Exponential relationships 281 Self-test 285

17 CIRCULAR FUNCTIONS 28717.1 Angles of any magnitude 28717.2 The reciprocal ratios 29617.3 The co-ratios 29817.4 Circular measure 29817.5 Angular velocity (rotational speed) 30017.6 The circle: length of arc, area of sector, area of

segment 301 Self-test 302

18 TRIGONOMETRIC FUNCTIONS AND PHASE ANGLES 303

18.1 Graphs of a sin bu, a cos bu 30318.2 The tangent curves 30818.3 Sketch-graphs of a sin (u 6 a), a cos (u 6 a) 30818.4 The functions a sin (bu 1 a), a cos (bu 1 a) 31018.5 Phase angles 31418.6 Adding sine and cosine functions 316 Self-test 318

19 TRIGONOMETRY OF OBLIQUE TRIANGLES 319

19.1 The sine rule 31919.2 The ambiguous case 32219.3 The cosine rule 32319.4 The use of the sine and cosine rules 326 Self-test 328

20 TRIGONOMETRIC IDENTITIES 33020.1 Definition 330

20.2 Trigonometric identities 33020.3 Proving an identity 33220.4 Summary: trigonometric identities 33320.5 Other trigonometric identities 336 Self-test 338

21 INTRODUCTION TO VECTORS 34021.1 A mathematical vector 34021.2 Scalar and vector quantities 34021.3 Addition of vectors and vector quantities 34121.4 Analytical addition of two vectors at right

angles to each other 34321.5 Resolution of a vector into two components at

right angles 34421.6 Resultant of a number of vectors analytically 34821.7 Addition of vectors graphically 34921.8 Equilibrium 352 Self-test 355

22 ROTATIONAL EQUILIBRIUM AND FRAME ANALYSIS 357

22.1 A few reminders 35722.2 The moment of a force about a point 35822.3 An introduction to the analysis of frames 361

23 DETERMINANTS AND MATRICES 36723.1 Definition and evaluation of a 2 3 2

determinant 36723.2 Solution of simultaneous equations using 2 3 2

determinants 36823.3 Matrices: introduction 37023.4 Some definitions and laws 37223.5 Multiplication of matrices 37323.6 Compatibility 37623.7 The identity matrix, I 37723.8 The inverse matrix, A21 37923.9 The algebra of matrices 38123.10 Expressing simultaneous equations in matrix

form 38423.11 Solving simultaneous linear equations using

matrices 38523.12 3 3 3 Determinants: definition and

evaluation 39423.13 Solutions of simultaneous linear equations

using 3 3 3 determinants 396

24 STATISTICS AND PROBABILITY 40124.1 Statistics 40124.2 Median, mode and mean 40124.3 Exploring the extremes of the data 40324.4 Exploring the location of the data 40524.5 Graphs 40924.6 Probability 41024.7 Mean and standard deviation of binomial

events 41324.8 Normal curve 415 Self-test 421

Appendix A 423Appendix B 425Answers 426Index 457

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17.5 Angular velocity (rotational speed) 300

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17.5 Angular velocity (rotational speed) 30017.6 The circle: length of arc, area of sector, area of

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17.6 The circle: length of arc, area of sector, area of

18 TRIGONOMETRIC FUNCTIONS AND PHASE

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18 TRIGONOMETRIC FUNCTIONS AND PHASE

a

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a cos

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cos b

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u bu b

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18.2 The tangent curves 308Sample

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23.2 Solution of simultaneous equations using 2

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23.2 Solution of simultaneous equations using 2 determinants 368

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determinants 36823.3 Matrices: introduction 370

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23.3 Matrices: introduction 37023.4 Some definitions and laws 372

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23.4 Some definitions and laws 37223.5 Multiplication of matrices 373

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23.5 Multiplication of matrices 37323.6 Compatibility 376

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23.6 Compatibility 37623.7 The identity matrix, I 377

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23.7 The identity matrix, I 37723.8 The inverse matrix, A

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23.8 The inverse matrix, A23.9 The algebra of matrices 381

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23.9 The algebra of matrices 381

only22 ROTATIONAL EQUILIBRIUM AND FRAME

only22 ROTATIONAL EQUILIBRIUM AND FRAME

ANALYSIS 357

onlyANALYSIS 357

22.1 A few reminders 357

only22.1 A few reminders 357

22.2 The moment of a force about a point 358

only22.2 The moment of a force about a point 358

22.3 An introduction to the analysis of frames 361

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23 DETERMINANTS AND MATRICES 367only

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About the authorsVIII

ABOUT THE AUTHORS

ACKNOWLEDGMENTS

Vincent Kelly, BEng (Civil), is currently Senior Civil Engineer at Hydro Tasmania. He is an industry

expert with more than 15 years’ engineering experience and brings this strong practical knowledge

to the text. His particular areas of interest are building design and renewable energy.

Blair Alldis BA, BSc, the original author of the previous six successful editions of Mathematics

for Technicians, has co-authored this seventh edition with Vincent Kelly. Blair has had extensive

experience in teaching mathematics to engineering students at TAFE and was Senior Head Teacher

of Mathematics at Sydney Institute of TAFE, Randwick College, prior to retiring.

Many thanks to Blair Alldis for the privilege of working on this book; to my wife Corin for all

her support and assistance; and to my sons Zac and Sam for their never-ending distraction and

amusement. Thanks also to the teachers who assisted in the development of this new edition with

their responses to the review questionnaire: Barry Russell from Skills Tech Eagle Farm, Queensland;

Les Smith from South Western Sydney Institute of TAFE; Granville College; and John Hungerford.

Finally, thanks to the team at McGraw-Hill who have been involved in the production of this edition,

in particular Norma Angeloni Tomaras (publisher), Lea Dawson (development editor), Yani Silvana

(senior production editor) and Ross Blackwood (freelance copyeditor).

Vincent Kelly

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Les Smith from South Western Sydney Institute of TAFE; Granville College; and John Hungerford. Sample

Les Smith from South Western Sydney Institute of TAFE; Granville College; and John Hungerford.

Finally, thanks to the team at McGraw-Hill who have been involved in the production of this edition, Sample

Finally, thanks to the team at McGraw-Hill who have been involved in the production of this edition, Sample

in particular Norma Angeloni Tomaras (publisher), Lea Dawson (development editor), Yani Silvana Sample

in particular Norma Angeloni Tomaras (publisher), Lea Dawson (development editor), Yani Silvana

(senior production editor) and Ross Blackwood (freelance copyeditor). Sample

(senior production editor) and Ross Blackwood (freelance copyeditor).

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Preface IX

PREFACE Mathematics for Technicians was developed to motivate engineering students to see the relevance of

mathematics to their future careers. For this seventh edition, Vincent Kelly uses his extensive practical

engineering experience to build on the strengths of this essential and inspiring well-known text.

Practical topics and examples will help to equip students with a sound working knowledge of the

elementary mathematics they require to succeed in the many engineering projects they will work on.

The author has also aligned the text with the requirements of TAFE engineering subject units. To

achieve this, the following changes have been made: ■ Chapters 17 and 18 have been revised. Chapter 17 now concentrates on circular functions

and Chapter 18 focuses on trigonometric functions and phase angles. ■ The material on determinants and matrices in Chapter 23 has been extended to more thoroughly

cover the solution of simultaneous linear equations using matrices, including the solution of

three equations with three unknowns. ■ A new chapter (Chapter 24) on statistics and probability has been added. ■ Supplementary material previously supplied on a CD is now available on the Online Learning

Centre (OLC) and includes additional exercises.

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Chapters 17 and 18 have been revised. Chapter 17 now concentrates on circular functions

only

Chapters 17 and 18 have been revised. Chapter 17 now concentrates on circular functions

The material on determinants and matrices in Chapter 23 has been extended to more thoroughly

onlyThe material on determinants and matrices in Chapter 23 has been extended to more thoroughly

cover the solution of simultaneous linear equations using matrices, including the solution of

onlycover the solution of simultaneous linear equations using matrices, including the solution of

A new chapter (Chapter 24) on statistics and probability has been added. only

A new chapter (Chapter 24) on statistics and probability has been added. only

Supplementary material previously supplied on a CD is now available on the Online Learning only

Supplementary material previously supplied on a CD is now available on the Online Learning

Text at a glanceX

TEXT AT A GLANCE Setting a clear agenda Each chapter starts by clearly

defining learning objectives.

Theory is covered clearly

throughout each topic. Where

appropriate, mathematical

rules, advice, notes and

definitions are highlighted

separately for added emphasis.

Worked examples are

provided to illustrate the

theory and provide a template

for students to develop their

own skills.

Practice makes perfect To further develop students’

skills and reinforce

understanding, several

levels of testing are provided

throughout the text. A set of

exercises follows each theory

section. The symbol

denotes more difficult exercises.

I f y o u h a v e d if f ic u l t y w it h e x e r c is e s in t h is c h a p t e r , y o u s h o u l d c o n s u l t y o u r t e a c h e r b e f o r e c o n t in u in g w it h t h is c o u r s e a n d s e e k r e m e d ia l a s s is t a n c e .

Learning objectives O n c o m p l e t io n o f t h is c h a p t e r y o u s h o u l d b e a b l e t o j u d g e w h e t h e r y o u r k n o w l e d g e o f t h e f o l l o w in g t o p ic s is s a t is f a c t o r y a s a b a s is f o r t h is c o u r s e . ■ S o l v e p r o b l e m s i n v o l v i n g r a t i o s . ■ S o l v e p r o b l e m s i n v o l v i n g d i r e c t v a r i a t i o n ( p r o p o r t i o n ) , i n v e r s e v a r i a t i o n a n d j o i n t v a r i a t i o n . ■ U s e p e r c e n t a g e t o s o l v e p r o b l e m s m e n t a l l y , m a n u a l l y a n d u s i n g a c a l c u l a t o r . ■ P e r f o r m c o n v e r s i o n s b e t w e e n f r a c t i o n s , d e c i m a l s a n d p e r c e n t a g e s . ■ A p p l y y o u r k n o w l e d g e o f r a t i o , p r o p o r t i o n a n d p e r c e n t a g e t o s o l v e p r a c t i c a l p r o b l e m s .

Rule: When an equation contains fractions, it is usually advisable to clear all fractions first by multiplying both sides by the LCM of the denominators.

Rule: Since in an equation the pronumeral is standing for some definite number (unknown until the equation is solved), we can multiply both sides of an equation by the pronumeral.

Examples 1 I f A 5 3 t 2 2 a n d B 5 5 2 t, e x p r e s s 2 A 2 3 B in t e r m s o f t.

Solution 2 A 2 3 B 5 2 ( 3 t 2 2 ) 2 3 ( 5 2 t )

5 6 t 2 4 2 1 5 1 3 t5 9 t 2 1 9

2 I f x 53 2 k

2 a n d y 5

k 2 23

, e x p r e s s 2 x 2 6 y in t e r m s o f k.

Solution

2 x 2 6 y 5 2 a 3 2 k2

b 2 6 ak 2 23

b5 ( 3 2 k ) 2 2 ( k 2 2 )5 3 2 k 2 2 k 1 45 7 2 3 k

Exercises 4.8 1 Simplify:

a 5x 26x 2 4

2 b T 2

6T 2 9

3 c 1 2

8 1 6R

2

d R 1 3 26R 2 8

2 e 3V 2

2V 2 6

2 f 1 2 4t 2

8t 1 4

4

2 Express as single fractions:

a x

22

x 1 1

3 b

3

42

2 2 4E

3 c

2

32

3C 2 1

6

d 3V

42

2 1 3V

6 e

5

62

3L 1 5

4 f

4k

92

k 2 5

6

3 E i l f i

Treatment of theory

alL70772_fm_i-xii.indd x 9/5/12 6:03 PM

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Text at a glance XI

SELF-TEST 1 $272 is divided between two people so that the amounts they receive are in the ratio 3 : 5. How

much does each receive?

2 In a test, a student scores 67 marks out of a maximum possible mark of 82. What is the

approximate percentage mark?

3 The frequency of vibration of a stretched wire when plucked varies inversely as its length and

directly as the square root of its tension. If a wire that is 1.40 m long vibrates with a frequency of

376 per second (376 Hz), when its tension is 125 N:

a what will be its frequency if its length is 2.20 m and its tension is 234 N?

b what will be the approximate percentage increase or decrease in the frequency?

4 Evaluate mentally:

a 37% of 8 mL b 12% of $23.50

5 A 50 m cable extends by 25 mm under a load of 1 t. What is the percentage increase in its

length? (1000 mm 5 1 m)

6 The introduction of an extra machine into a workshop increases the production rate from 170 to

204 articles per week. What is the percentage increase in the production rate? 1 3

ANSWERS 1 Fractions and decimals Exercises 1.1 1 a 2 8 b 4 c 2 2 d 2 2 e 2 4 f 2 2 a 2 3

b 10 c 2 3 3 a 0 b 0 c 2 2 4 a 2 2 8 C

b 2 6 8 C 5 a contraction of 3 mm b contraction

of 5 mm 6 a loss of 2 dB b loss of 8 dB c gain

of 4 dB 7 a 2 b 14 8 a 2 6 b 2 8 c 2 4 d 4

e 6 f 12 9 a 4 b 2 2 c 6 d 2 4 e 2 8 f 2 1

Exercises 1.2 1 a 14 b 2 1 c 26 2 a 12 b 7 c 29 3 a 10

b 5 4 a 40 b 6 c 9 d 2

Exercises 1.3 1 a 2, 3, 5 b 3, 9 c 2, 3, 4 d 3, 5, 9 e 2, 3, 9

f 3, 5 g 2, 4 h 3, 9 2 a 1315 b 16

17 c 34 d 719

e 34 f 47 g 23 h 35 3 a 813 b 35 c 997

7000 d 4010 4 21 b 42 11 7 d 41 41 5 7

d 1.8 e 0.6 f 0.000 06 8 a 0.09 b 0.0004

c 0.36 d 1.44 e 0.0144 f 0.000 121 g 0.0009

h 1.21 9 a 0.2 b 0.9 c 0.07 d 1.2 10 a 26.08

b 1.203 c 230.7 d 0.0521 11 a 2.03 b 732.8

c 6.003 d 1.703 e 0.0732 f 0.0039 12 a 0.68

b 5.33 c 62.15 d 234.20 e 0.04 f 0.00 g 0.01

h 0.00 13 a 400 b 0.567 c 0.234 14 a 30

b 0.4 c 0.2 d 0.03 15 a 20 b 20 c 0.3

16 a 3.6 V b 200 m 17 a 0.12 V b 0.16 V

c 0.024 V d 0.48 V 18 a 150 b 0.06 s

19 a 1.46 mm b 43.8 mm c 58.4 mm d 7.3 mm

20 3066 mm 2 21 3.4 m/s 22 4000 W 23 19.8 J

24 20 s 25 a 1.5 s b 45 s 26 4.8 min

27 a 7000 kg b 0.002 m 3

Exercises 1.5 1 a 2500 b 2400 c 2500 d 600 e 700 f 800

To ensure complete

understanding of each chapter,

self-test questions that address

core concepts are provided

at the end of most chapters.

These tests are an excellent

revision tool for the classroom.

Answers to all exercises and

self-test questions are provided

at the back of the book and

online, enabling self-paced

learning.

alL70772_fm_i-xii.indd xi 9/5/12 6:03 PM

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only

only

A 50 m cable extends by 25 mm under a load of 1 t. What is the percentage increase in its

only

A 50 m cable extends by 25 mm under a load of 1 t. What is the percentage increase in its

The introduction of an extra machine into a workshop increases the production rate from 170 to

only

The introduction of an extra machine into a workshop increases the production rate from 170 to

204 articles per week. What is the percentage increase in the production rate?

only204 articles per week. What is the percentage increase in the production rate?

only

only

0.0004 only

0.0004

0.0009 only

0.0009

Answers to all exercises and

only Answers to all exercises and

self-test questions are provided

onlyself-test questions are provided

at the back of the book and only

at the back of the book and

online, enabling self-paced only

online, enabling self-paced

learning. only

learning.

E-student / E-instructorXII

E-STUDENT / E-INSTRUCTOR E-student

The Online Learning Centre (OLC) that accompanies this text helps you get the most from your

course. It provides a powerful learning experience beyond the printed page.

www.mhhe.com/au/alldis7e

Supplementary material Supplementary material is now included on the OLC. It provides answers to the questions within

the text, as well as additional exercises and theories on a wide range of content. Within the text, the

OLC symbol indicates that supplementary material is available to enhance your learning

experience.

E-instructor Instructors have additional password-protected access to an instructor-specific area, which includes

the following resources.

Teaching notes To assist in classroom preparation, instructors will have access to teaching notes on selected

chapters.

EZ Test Online EZ Test Online is a powerful and easy-to-use test generator for creating paper or digital tests. It

allows easy ‘one click’ export to course management systems such as Blackboard and straightforward

integration with Moodle.

Testbank A bank of test questions written specifically for this text allows instructors to build examinations and

assessments quickly and easily. The testbank is available in a range of flexible formats: in Microsoft

Word ® , in EZ Test Online or formatted for delivery via Blackboard or other learning-management

systems.

alL70772_fm_i-xii.indd xii 9/5/12 6:03 PM

Sample

Instructors have additional password-protected access to an instructor-specific area, which includes

Sample


To assist in classroom preparation, instructors will have access to teaching notes on selected

Sample

To assist in classroom preparation, instructors will have access to teaching notes on selected

EZ Test Online is a powerful and easy-to-use test generator for creating paper or digital tests. It

Sample

EZ Test Online is a powerful and easy-to-use test generator for creating paper or digital tests. It

Sample

allows easy ‘one click’ export to course management systems such as Blackboard and straightforward Sample

allows easy ‘one click’ export to course management systems such as Blackboard and straightforward

integration with Moodle. Sample

integration with Moodle.

only Supplementary material is now included on the OLC. It provides answers to the questions within

only Supplementary material is now included on the OLC. It provides answers to the questions within

the text, as well as additional exercises and theories on a wide range of content. Within the text, the

onlythe text, as well as additional exercises and theories on a wide range of content. Within the text, the

indicates that supplementary material is available to enhance your learning

only indicates that supplementary material is available to enhance your learning

Instructors have additional password-protected access to an instructor-specific area, which includes only


CHAPTER 16

16.1 COMPOUND INTEREST To understand the material in this chapter, you need to apply the knowledge from topics covered in

chapter 2 (Section 2.6 Percentages ) and chapter 14 (Section 14.9 Exponential equations ).

If a number N is increased by r %, it becomes N 1 a r

1003 Nb 5 N 3 a1 1

r

100b.

If this result is increased by r %, it becomes N 3 a1 1r

100b 3 a1 1

r

100b.

If this ‘compounding’ process is performed n times, N increases to:

N 3 c a1 1r

100b 3 a1 1

r

100b 3 a1 1

r

100b 3 cn terms d 5 N 3 a1 1

r

100b

n

.

If money is invested in such a way that after certain regular time periods the interest is calculated

and added to the account so that it attracts interest during the next period, the interest is said to be

‘compounded’.

If $ P (the ‘principal’) is invested for n periods at a compound interest rate of r % per period, there

will be n interest payments into the account, each of r% of the amount in the account at that time.

The amount to which the money grows in these n periods is given by:

COMPOUND INTEREST: EXPONENTIAL GROWTH AND DECAY Learning objectives ■ Solve compound interest problems. ■ Draw graphs relating to exponential growth and decay. ■ Solve exponential growth and decay problems.

Rule: A 5 P 3 a1 1r

100b

n

alL70772_ch16_273-286.indd 273 23/08/12 10:30 AM

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To understand the material in this chapter, you need to apply the knowledge from topics covered in

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To understand the material in this chapter, you need to apply the knowledge from topics covered in


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aSample

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Draw graphs relating to exponential growth and decay. only

Draw graphs relating to exponential growth and decay.

Part 3 | Applied mathematics274

16.2 EXPONENTIAL GROWTH The more frequently the interest is calculated and compounded, the more rapidly the amount in the

account grows.

Example Calculate the amount that $5000 grows to when it is invested for three weeks at an interest rate of 13% per year, the interest being compounded daily.

Solution Interest rate 5 13% per year 5

13365 % per day, and there will be 21 interest payments into the account.

6 A 5 $5000 3 a1 1

13365

100b

21

< $5037.53

H

Exercises 16.1 1 Money is invested at an interest rate of 6% p.a. (per year), the interest being paid and

compounded at the end of each year.

a By what percentage does the investment increase in:

i 3 years? ii 8 years?

b For how many complete years must the money remain invested before it has more than

doubled?

2 The sum of $2550 is invested at an interest rate of 0.7% per month, the interest being paid and

compounded at the end of each month.

a State the exact amount on deposit at the end of 5 months and also its value correct to the

nearest cent.

b By what percentage will the investment increase in 1 year? State the exact percentage and also

its value correct to 3 significant figures.

3 $3000 is invested for 26 weeks at an interest rate of 7% p.a., the interest being compounded

daily. How much total interest is paid on this investment?

This is the ‘compound interest formula’ where:

P is the principal sum invested

r% is the interest rate over the period of time between interest payments

A is the amount to which the principal grows after a time interval of n of the above periods.

alL70772_ch16_273-286.indd 274 23/08/12 10:30 AM

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Compound interest: exponential growth and decay | Chapter 16 275

Example Calculate the amount that $10 000 grows to when it is deposited for 3 years at an interest rate of 5% per year.

Solution The amount A to which the principal grows in the three years is given by:

■ if interest is compounded yearly: A 5 $10 000 3 a1 15

100b3

5 $11 576.25

■ if interest is compounded monthly: A 5 $10 000 3 a1 1

512

100b

36

5 $11 614.72

■ if interest is compounded weekly: A 5 $10 000 3 a1 1

552

100b

3352

5 $11 617.51

■ if interest is compounded daily: A 5 $10 000 3 a1 1

5365

100b

33365

5 $11 618.22.

It can be shown that as n becomes larger and larger (interest compounded each hour, minute,

second, . . .), the amount A does not increase indefinitely (i.e. ‘without bound’) but approaches a

limit, which is given by

Rule: A 5 P 3 ekt

where

k is the rate of interest per year expressed as a decimal

t is the investment period in years

e is a constant named after the Swiss mathematician Leonard Euler.

H Note: The constant e is an irrational number—that is, a non-recurring, non-repeating decimal (like p ). Its approximate value is 2.718. This number is so important in technology that, like p , all scientific calculators provide an e x key. You should check on your calculator that e 1 ( 5 e ) < 2.718. You should be able to obtain, for example, e 3 < 20.09 and e 2 5 < 6.738 3 10 2 3 .

Note: Students who are interested should study Appendix A, which contains more information concerning the constant e and the derivation of the above formula for exponential growth.

You can now find the limit to which the amount in the previous example approaches—the amount if

the interest is paid continuously:

A 5 P 3 ekt 5 $10 000 3 e0.0533 5 $10 000 3 e0.15 5 $11 618.34

Regardless of how often the interest is compounded, the total amount can never exceed this amount.

This is the amount if the interest is paid continuously. Banks pay interest on some accounts daily

or continuously and in both cases the continuous formula would be used since there is very little

difference between the amounts in each case.

alL70772_ch16_273-286.indd 275 23/08/12 10:30 AM

Sample

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5

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5 P

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P 3

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3 e

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ekt

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kt

is the rate of interest per year expressed as a decimal

Sample

is the rate of interest per year expressed as a decimal

is the investment period in years

Sample

is the investment period in years

is a constant named after the

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is a constant named after the Sw

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Swiss mathematician Leonard Euler.

Sample

iss mathematician Leonard Euler.

Sample

is an irrational number—that is, a non-recurring, non-repeating decimal (like

Sample

is an irrational number—that is, a non-recurring, non-repeating decimal (like

approximate value is 2.718. This number is so important in technology that, like

Sample

approximate value is 2.718. This number is so important in technology that, like

key. You should check on your calculator that Sample

key. You should check on your calculator that 20.09 and Sam

ple

20.09 and eSample

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2Sample

2 5 Sample

5 <Sample

< 6.738 Sample

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10 Sample

Students who are interested should study Appendix A, which contains more information concerning Sam

ple

Students who are interested should study Appendix A, which contains more information concerning

only

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only5 $11 618.22.

only$11 618.22.

becomes larger and larger (interest compounded each hour, minute,

only

becomes larger and larger (interest compounded each hour, minute,

(i.e. ‘without bound’) but approaches a only

(i.e. ‘without bound’) but approaches a


Actually all growth must be in ‘spurts’ but these are often so small and so frequent (e.g. molecule by

molecule, quantum by quantum, cent by cent) that the growth may be considered to be continuous.

Note: The above formula for exponential growth applies, of course, not only to a quantity of money but to any quantity that grows in this continuous manner where the percentage rate of growth is constant and hence the actual rate of growth is proportional to the amount present at any instant. With a bank deposit invested at compound interest paid continuously, the percentage rate of growth remains constant and so the actual rate of growth (e.g. in dollars per day) increases in proportion to the amount of money in the account.

This type of growth is quite common, an example being population growth (e.g. bacterial, animal or

human). When the population of rabbits doubles, the rate of breeding (e.g. in rabbits per second)

will double; the rate of growth of the population is proportional to the size of the population at

every particular instant. The growth of the population can be regarded as continuous only when

the population is large. When the population is small (as was the original population of rabbits

in Australia), the growth could not be regarded as continuous. All runaway chain reactions are

examples of this type of growth, such as occurs in uncontrolled nuclear fission. This type of growth

is sometimes referred to as ‘snowballing’ because when a snowball rolls down a snow-covered slope,

the rate at which it picks up snow at any instant (e.g. in grams per second) is proportional to the

weight of the snowball at that instant.

SUMMARY ■ Exponential growth is defined as continuous growth in which the actual rate of growth at any instant

is proportional to the quantity present at that instant.

■ The formula for exponential growth is Q 5 Q 0 e kt ,

where: Q0 is the quantity originally present (i.e. the value of Q when t 5 0)k is the percentage continuous rate of growth during a specifi ed period of time, expressed as a decimalQ is the quantity present at the end of t of the above time periods. H

Exercises 16.2 1 If $ P is invested at a compound interest rate of 8% p.a., state the exact amount of money on

deposit at the end of 1 year:

a if the interest is paid at the end of each year

b if the interest is paid quarterly

c if the interest is paid continuously

2 If $10 000 is invested at an interest rate of 4% p.a., what will this amount become at the end of

1 year, to the nearest cent:

a if interest is paid each year? b if interest is paid each quarter?

c if interest is paid each month? d if interest is paid each week?

e if interest is paid each day? f if interest is paid continuously?

alL70772_ch16_273-286.indd 276 23/08/12 10:30 AM

Sample

Sample

The above formula for exponential growth applies, of course, not only to a quantity of money but

Sample

The above formula for exponential growth applies, of course, not only to a quantity of money but to any quantity that grows in this continuous manner where the percentage rate of growth is constant and

Sample

to any quantity that grows in this continuous manner where the percentage rate of growth is constant and hence the actual rate of growth is proportional to the amount present at any instant. With a bank deposit

Sample

hence the actual rate of growth is proportional to the amount present at any instant. With a bank deposit invested at compound interest paid continuously, the percentage rate of growth remains constant and so the

Sample

invested at compound interest paid continuously, the percentage rate of growth remains constant and so the actual rate of growth (e.g. in dollars per day) increases in proportion to the amount of money in the account.

Sample

actual rate of growth (e.g. in dollars per day) increases in proportion to the amount of money in the account.


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the population is large. When the population is small (as was the original population of rabbits Sample

the population is large. When the population is small (as was the original population of rabbits

in Australia), the growth could not be regarded as continuous. All runaway chain reactions are Sample

in Australia), the growth could not be regarded as continuous. All runaway chain reactions are

examples of this type of growth, such as occurs in uncontrolled nuclear fission. This type of growth Sample

examples of this type of growth, such as occurs in uncontrolled nuclear fission. This type of growth Sample

is sometimes referred to as ‘snowballing’ because when a snowball rolls down a snow-covered slope, Sample

is sometimes referred to as ‘snowballing’ because when a snowball rolls down a snow-covered slope,

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only growth must be in ‘spurts’ but these are often so small and so frequent (e.g. molecule by

only growth must be in ‘spurts’ but these are often so small and so frequent (e.g. molecule by

molecule, quantum by quantum, cent by cent) that the growth may be only

molecule, quantum by quantum, cent by cent) that the growth may be consideredonly

consideredonly

The above formula for exponential growth applies, of course, not only to a quantity of money but only

The above formula for exponential growth applies, of course, not only to a quantity of money but to any quantity that grows in this continuous manner where the percentage rate of growth is constant and on

lyto any quantity that grows in this continuous manner where the percentage rate of growth is constant and

if interest is paid each quarter?

only

if interest is paid each quarter?

if interest is paid each week?

only

if interest is paid each week? if interest is paid each week?

only


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if interest is paid each week?

if interest is paid continuously?

only if interest is paid continuously? if interest is paid continuously?


























only if interest is paid continuously?


Example If the mass of a culture increases exponentially from an original mass of 3.45 g at a continuous rate of growth of 13% per hour, find the mass present after two days of growth.

Solution k 5 13% 5 0.13 per hour

t 5 2 3 24 5 48 hours

Q 5 Q0ekt

5 3.45 3 e0.13348

5 3.45 3 e6.24

< 1770 g We could equally well have used 1 day as the time period, in which case k 5 0.13 3 24 5 3.12 per day, and t 5 2 days and e kt 5 e 6.24 as before.

Note: When there is an exponential decrease (‘decay’) the value of k is negative.

Exercises 16.3 1 The radioactive element radon has a half-life of 3.82 days. (The half-life is the time taken for half

of any mass to disintegrate.)

Find:

a the value of k, the continuous fractional rate of growth

b how long it will take for 100 g of radon to decay to 45.0 g.

2 The population of a particular town, which is currently 10 400, is estimated to be increasing at a

rate of 7% per year. Assuming that this rate of growth continues:

a what is the estimate of the population in 5 years’ time?

b how long will it take for the population to double?

c how long will it take for the population to increase by 35%?

3 When the key in this circuit is closed, the current decreases

exponentially according to the formula i 5E

R3 e2

1RC t A/s.

If E 5 180 V, C 5 120 m F and R 5 3.8 k V , find the current:

a immediately after the key is closed

b 730 ms after the key is closed.

4 A manufacturer of yeast finds that their culture grows

exponentially at the rate of 13% per hour.

a What mass will 3.7 g of yeast grow to:

i in 7.0 hours? ii in two days?

R

C

E

continued

alL70772_ch16_273-286.indd 277 23/08/12 10:30 AM

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The radioactive element radon has a half-life of 3.82 days. (The

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The radioactive element radon has a half-life of 3.82 days. (The The radioactive element radon has a half-life of 3.82 days. (The

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The radioactive element radon has a half-life of 3.82 days. (The

the continuous fractional rate of growth

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the continuous fractional rate of growth the continuous fractional rate of growth

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the continuous fractional rate of growth

how long it will take for 100 g of radon to decay to 45.0 g.

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how long it will take for 100 g of radon to decay to 45.0 g. how long it will take for 100 g of radon to decay to 45.0 g.

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how long it will take for 100 g of radon to decay to 45.0 g.

The population of a particular town, which is currently 10 400, is estimated to be increasing at a

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The population of a particular town, which is currently 10 400, is estimated to be increasing at a The population of a particular town, which is currently 10 400, is estimated to be increasing at a

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how long will it take for the population to double? Sample

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how long will it take for the population to double?

how long will it take for the population to increase by 35%? Sample

how long will it take for the population to increase by 35%? how long will it take for the population to increase by 35%? Sample



















how long will it take for the population to increase by 35%?

When the key in this circuit is closed, the current decreases Sample

When the key in this circuit is closed, the current decreases When the key in this circuit is closed, the current decreases Sample







When the key in this circuit is closed, the current decreases

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5 3.12 per day, and

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only When there is an exponential decrease (‘decay’) the value of

only When there is an exponential decrease (‘decay’) the value of k

onlyk is negative.

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half-lifeon

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lyhalf-life


16.3 GRAPHS OF EXPONENTIAL FUNCTIONS Functions in which the variable appears as an exponent (index) are called ‘exponential’ functions.

Examples are 3.2 x , 7.15 2 3 x , 2.86 4.31 x 2 2.45 . The general expression for an exponential function is

K 3 a bx 1 c , where each of the numbers K, a, b and c may be positive or negative and c may be zero.

A very important property of these functions is that equal increases in the value of x result in

equal percentage increases in the value of the function.

Nx1k 5 Nx 1 Nk Successive additions of k to the exponent result in a succession of

Nx12k 5 Nx1k 1 Nk multiplications by the same number, N k . Hence, as shown in

Nx13k 5 Nx12k 1 Nk chapter 2 (section 2.6, Percentages ) successive additions of k to the

Nx14k 5 Nx13k 1 Nk exponent result in a succession of increases by the same percentage.

Consider the function y 5 K 3 a bx 1 c . If x is increased by t, the value of y becomes

K 3 a b ( x 1 t ) 1 c 5 K 3 a bx 1 c 1 bt 5 K 3 a bx 1 c 3 a bt . Hence the value of y is doubled when a bt 5 2,

i.e. when bt log a 5 log 2, or t 5log 2

b log a. There is no need to memorize this result but it will assist

you to understand the following work if you can follow the reasoning.

continued

b What mass must be present initially in order to acquire a mass of 15 kg at the end of two

days?

c How long will it take for any given mass of this yeast to double?

5 In the inversion of raw sugar, at any particular moment the rate of decrease of the mass of sugar

present is proportional to the mass of sugar remaining. (Remember that this means that the rate

of decrease is exponential. ) If after 8 hours 53 kg has reduced to 37 kg, how much sugar will

remain after a further 24 hours?

y

x

To sketch any curve of the form y 5 K 3 a bx 1 c

when no range of required values is specified for x

or y, you can either set up a table of corresponding

values of x and y as described in chapter 9 (section 9.6

Dependent and independent variables ) or use the

following general method, which is much quicker and

also provides an understanding of the properties of

exponential functions.

alL70772_ch16_273-286.indd 278 23/08/12 10:30 AM

Sample

A very important property of these functions is that equal increases in the value of

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increases in the value of the function.

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increases in the value of the function.

Successive additions of

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Successive additions of k

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k to the exponent result in a succession of

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to the exponent result in a succession of

multiplications by the same number,

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multiplications by the same number,

chapter 2 (section 2.6, Percentages ) successive additions of

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chapter 2 (section 2.6, Percentages ) successive additions of

exponent result in a succession of increases by the same percentage.

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exponent result in a succession of increases by the same percentage.

y

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only Functions in which the variable appears as an exponent (index) are called ‘exponential’ functions.

. The general expression for an exponential function is

only . The general expression for an exponential function is

may be positive or negative and only

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A very important property of these functions is that equal increases in the value ofonly



1 Draw axes as shown, placing 2 equally spaced marks on the x -axis on each side of the origin and

place 5 marks on the y -axis starting with the top one whose position is the highest you wish your

graph to be and each of the succeeding marks below being at half the remaining distance down

to the origin. This results in 5 marks, each one being twice the distance from the origin as the

mark below it.

2 Calculate the value of y when x 5 0 and label this value on the middle of your 5 marks on the

y -axis. Then mark the values on the other 4 marks on the y -axis, each value being double the

value on the mark below it.

3 Calculate the amount by which x must increase in order to double any y -value. Place this value

next to the first mark to the right of the origin on the x -axis, and place corresponding values next

to the other marks.

4 Mark 5 points on the curve as shown on the figure above, and sketch the curve. To decide which

of the two curves below to draw, examine what happens to the value of y as x S . This will

depend on whether the exponent b is positive or negative.

y

x

y = K abx+c when b > 0as x �, y � as x –�, y 0

y = a bx+c when b < 0as x �, y 0as x –�, y �

y

x

All the above should become clear if you study the examples below. Note that in each case the axes

are drawn first, then the scale-marks made on each axis as described above.

Example Sketch the curve y 5 3.2 x ■ As x S , y S ; as x S 2 , y S 0 ■ When x 5 0, y 5 1 ■ y doubles when 3.2 x increases from 1 to 2.

If 3.2x 5 2

then log 3.2x 5 log 2

continued

alL70772_ch16_273-286.indd 279 23/08/12 10:30 AM

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All the above should become clear if you study the examples below. Note that in each case the axes

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are drawn first, then the scale-marks made on each axis as described above.

only

Mark 5 points on the curve as shown on the figure above, and sketch the curve. To decide which

only

Mark 5 points on the curve as shown on the figure above, and sketch the curve. To decide which

of the two curves below to draw, examine what happens to the value of

onlyof the two curves below to draw, examine what happens to the value of y

onlyyof the two curves below to draw, examine what happens to the value of yof the two curves below to draw, examine what happens to the value of

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continued

6 x log 3.2 5 log 2

6 x 5 ( log 2 ) 4 ( log 3.2 )

6 x < 0.6

We now have the scales for both axes: ■ On the x -axis the equally-spaced values 0.6 units apart. ■ On the y -axis the value doubles with each increase of 0.6

on the x -axis. ■ These 5 points are sufficient to show the shape

of the curve: ( 2 1.2, 0.25), ( 2 0.6, 0.5), (0, 1), (0.6, 2), (1.2, 4).

y = 3.2 x

y

x

4

2

1

0.50.25

–1.2 –0.6 0 0.6 1.2

Example Sketch the curve y 5 18 3 x 1 2 . ■ As x S , y S ; as x S 2 , y S 0. ■ When x 5 0, y 5 18 2 5 324. ■ y doubles when log 18 3 x 1 2 5 2 3 324 5 648

6 log 183x12 5 log 648

6 ( 3x 1 2 ) log 18 5 log 648

6 3x 1 2 5 ( log 648 ) 4 ( log 18 ) < 2.24

6 x < 0.08

We now have the scales for both axes: ■ On the x -axis the equally-spaced values 0.08 units apart. ■ On the y -axis the value doubles with each increase of

0.08 on the x -axis. ■ These 5 points are sufficient to show the shape of the curve:

( 2 0.16, 81), ( 2 0.08, 162), (0, 324), (0.08, 648), (0.16, 1296).

y

x

1296

162

-0.16 -0.08 0 0.08 0.16

648

324

81

y = 18 3x+2

Example Sketch the curve y 5 5.9 2 2 3 x . ■ As x S , y S 0; as x S 2 , y S ■ When x 5 0, y 5 5.9 2 < 34.8, ■ y doubles when 5.9 2 2 3 x increases to 69.6

6 ( 2 2 3x ) log 5.9 5 log 69.6

6 2 2 3x 5 ( log 69.6 ) 4 ( log 5.9 )

alL70772_ch16_273-286.indd 280 23/08/12 10:30 AM

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324

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We now have the scales for both axes:

-axis the equally-spaced values 0.08 units apart.

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-axis the equally-spaced values 0.08 units apart. -axis the value doubles with each increase of Sam

ple

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16.4 EXPONENTIAL RELATIONSHIPS Three very common relationships are:

6 2 2 3x < 2.39

6 3x < 20.39

6 x < 20.13

We now have the scales for both axes: ■ On the x -axis the equally-spaced values 0.13 units apart ■ On the y -axis the value doubles with each increase of

2 0.13 on the x -axis, i.e. with each decrease of 0.13. ■ These 5 points are sufficient to show the shape

of the curve:( 2 0.26, 13.9), ( 2 0.18, 69.6), (0, 34.8), (0.13, 17.4), (0.26, 8.7).

–0.26 –0.13 0 0.13 0.26

y = 5.9 2-3x

y

x

139

69.6

34.8

17.48.7

Exercises 16.4 Sketch each of the following curves. You may use the long method by setting up your own table of

values for x and y or the quicker method described in the examples above. The answers provided are

the coordinates of the 5 points obtained using the quick method, but if you use the long method the

5 points given in the answers should lie on your curve.

1 y 5 6.3 1.25 x 1 0.83 2 y 5 0.78 4.5 x 2 5.6 3 y 5 4.3 2.6 2 2.5 x

4 y 5 13 0.9 x 1 1.3 5 y 5 81 1.3 x 2 0.39

t

Q

Q0

Q = Q0.ekt

Q

t

Q0 Q = Q0.e–kt

t

Q

QL

Q = QL(1 – e–kt)

The values of Q, Q 0 , t and k were defined in the Summary in section 16.2 above on page 276.

a b c

alL70772_ch16_273-286.indd 281 23/08/12 10:30 AM

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16.4 EXPONENTIAL RELATIONSHIPS Sample

16.4 EXPONENTIAL RELATIONSHIPS Three very common relationships are:Sam

ple

Three very common relationships are:Sample

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Sketch each of the following curves. You may use the long method by setting up your own table of

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Sketch each of the following curves. You may use the long method by setting up your own table of Sketch each of the following curves. You may use the long method by setting up your own table of

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or the quicker method described in the examples above. The answers provided are

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or the quicker method described in the examples above. The answers provided are or the quicker method described in the examples above. The answers provided are

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the coordinates of the 5 points obtained using the quick method, but if you use the long method the the coordinates of the 5 points obtained using the quick method, but if you use the long method the

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5 points given in the answers should lie on your curve. 5 points given in the answers should lie on your curve.

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5 0.78

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Sketch each of the following curves. You may use the long method by setting up your own table of only

Sketch each of the following curves. You may use the long method by setting up your own table of Sketch each of the following curves. You may use the long method by setting up your own table of only



















or the quicker method described in the examples above. The answers provided are only

or the quicker method described in the examples above. The answers provided are or the quicker method described in the examples above. The answers provided are only












Examples of exponential growth See the above figure a for this example. ■ Population growth (human, animal, bacteria cultures—provided nothing such as famine interferes

with the rate of growth). ■ All ‘chain reactions’.

Examples of exponential decay See the above figure b for this example. ■ The radioactive decay of a substance, Q being the mass of the substance remaining unchanged at

time t. ■ A heated body cooling, Q being the excess of the body’s temperature above room temperature

( Newton’s law of cooling ). ■ The speed of a rotating flywheel when the power is disconnected, Q being measured in revolutions

per minute, for example. ■ Electrical quantities.

Examples of exponential growth towards an upper limit See the above figure c for this example. ■ The concentration of a substance during a chemical reaction, Q L being the final limiting

concentration. ■ The temperature of a body after it is placed in an oven, Q being the excess of the body’s temperature

above its original temperature, and Q L the temperature of the oven. ■ The speed of an electrically driven flywheel after the power is connected. ■ Electrical quantities.

Students of electrical engineering will easily be able to interpret the following diagrams. For a series

R–C or a series L-C circuit, when charging or discharging, all the currents and voltages increase or

decrease exponentially. (Current 5 rate of flow of charge.)

Charging Discharging

E

CR

E

CR

i 5E

R e2

1RC t

VR 5 iR 5 Ee21

RC t

VC 5 E 2 VR 5 E(1 2 e21

RC t

)

i 5E

R e2

1RC t

VR 5 iR 5 Ee21

RC t

VC 5 2VR 5 2Ee1

RC t

alL70772_ch16_273-286.indd 282 23/08/12 10:30 AM

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The temperature of a body after it is placed in an oven,

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The temperature of a body after it is placed in an oven, the temperature of the oven.

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the temperature of the oven. The speed of an electrically driven flywheel after the power is connected.

Sample

The speed of an electrically driven flywheel after the power is connected.


Sample


circuit, when charging or discharging, all the currents and voltages increase or

Sample

circuit, when charging or discharging, all the currents and voltages increase or

decrease exponentially. (Current

Sample

decrease exponentially. (Current

Sample

5

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5 rate of flow of charge.)

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rate of flow of charge.)


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C

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R

only

being measured in revolutions

only

being measured in revolutions

Examples of exponential growth towards an upper limit

only Examples of exponential growth towards an upper limit

The concentration of a substance during a chemical reaction, only

The concentration of a substance during a chemical reaction, Qonly

QLonly

L being the final limiting only

being the final limiting L being the final limiting Lonly

L being the final limiting L

being the excess of the body’s temperature only

being the excess of the body’s temperature the temperature of the oven.

only

the temperature of the oven.


Each of the above 12 equations is of the form Q 5 Q 0 e 2 kt or Q 5 Q L (1 2 e 2 kt ), whose graphs are

shown at the beginning of this section. In each case, quantity i, V R , V C or V L either commences at

an initial value Q 0 and decays towards zero, or commences at value zero and grows exponentially

towards a final limiting value, Q L .

For example, I 5E

R e2

1RC t means that the current commences at value

E

R (when t 5 0) and then

decays exponentially with time towards zero as t S . In this case the continuous fractional rate of

increase of i per unit of time is k 5 21

RC.

Also, i 5E

R (1 2 e2

RL t

) means that the current commences at value zero (when t 5 0) and grows

exponentially towards a final limiting value of E

R (as t S ). In this case the continuous fractional

rate of increase of i per unit of time is k 5R

L.

Note how the values of the functions change with equal increases in the value of x:

E

LR

E

LR

k

Q

4Q0

Exponential growth

2Q0

Q0

Q = Q0kt

t

Q Q = Q0e–kt

Q0

Q0

Q014

12

Exponential decayt

Q y = QL(1 – e–kt)

QL

QL

QL

34

12

Exponential growthfrom zero to a limit QL

The examples below show the steps in sketching such functions.

Rule: A reminder: log 10 10 5 1, log e e 5 1 (i.e. ln e 5 1)

i 5E

R (1 2 e2

RL

t )

VR 5 iR 5 E(1 2 e2RL t )

VL 5 E 2 VR 5 Ee2RL t

i 5E

R e2

RL t

VR 5 iR 5 Ee2RL t

VL 5 2VR 5 2Ee2RL t

alL70772_ch16_273-286.indd 283 23/08/12 10:30 AM

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means that the current commences at value

Sample

means that the current commences at value

S

Sample

S

Sample

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` . In this case the continuous fractional rate of

Sample

. In this case the continuous fractional rate of

) means that the current commences at value zero (when

Sample

) means that the current commences at value zero (when

ds a final limiting value of

Sample

ds a final limiting value of

E

Sample

E

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R

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R

(as

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(as t

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t S

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S `

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`

per unit of time is

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per unit of time is k

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k 5

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5

R

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R

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L

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L

.

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.

Note how the values of the functions change with equal increases in the value of

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Note how the values of the functions change with equal increases in the value of

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only ), whose graphs are

R

onlyR ,

only , R , R

onlyR , R V

onlyVV V

onlyV VC

onlyCVCV

onlyVCV or

only or C or C

onlyC or C V

onlyVL

onlyLVLV

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only either commences at L either commences at L

onlyL either commences at L

and decays towards zero, or commences at value zero and grows exponentially only

and decays towards zero, or commences at value zero and grows exponentially

means that the current commences at value only

means that the current commences at value Eonly

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Ee

only

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2

V

only

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Examples 1 Sketch the graph of V 5 10 e 0.02 t . When t 5 0, V 5 10 e 0.02 t 5 10 V doubles when 10 3 e 0.02 t 5 20, i.e. e 0.02 t 5 2, i.e. when 0.02 t 5 ln 2, or t < 35

V

t0

10

20

40 40

20

10

0

V

t

40

20

10

0

V

t35 70

2 Sketch the graph of M 5 400(1 2 e 2 1.25 t ). When t 5 0, M 5 400 (1 2 1) 5 0 As t S , M S 400 M halves when 400(1 2 e 2 1.25 t ) 5 200 i.e. when e 2 1.25 t 5 0.5 i.e. when 2 1.25 t 5 ln 0.5 or t < 0.56.

200

400

0 t

M

200

400

0 t

M

200

400

0 0.56 t

M

Exercises 16.5 1 Draw freehand, quick sketch-graphs of the functions:

a y 5 2 x , for 2 3 # x # 3 b y 5 2 2 x , for 2 3 # x # 3

c y 5 1 2 2 2 x , for 0 # x # 4 d y 5 327(1 2 2 2 x ), for 0 # x # 4

2 Draw accurate graphs of the functions:

a y 5 5 e 0.2 x for 2 6 # x # 10 b y 5 24(1 2 e 2 8 x ) for 0 # x # 0.5

3 a Calculate the values of the function Q 5 16(1 2 e 2 500 t ) when x has the values 2 2, 2 1, 0, 1, 2

and 3, and use these values to plot the graph of the function over this range.

alL70772_ch16_273-286.indd 284 23/08/12 10:30 AM

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laser

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AQ: Check


b What numbers correct to 2 significant figures would replace the numbers 1, 2 and 3 on the

x -axis of the graph in part a in order to convert it to the graph of the function:

i y 5 2 2 x ii y 5 16 x iii y 5 e x iv y 5 e 2 x ?

(Hint: Replace 1 by the number that makes the new function equal to 2.)

c What numbers would replace the numbers 1, 2, 4 and 8 on the y -axis of the graph in part a

in order to convert it to the graph of the function y 5 43 3 2 x ?

4 Find the approximate values of x for which:

a e x 5 2, 4, 8 and 16 b ex 5 1, 12, 14 and 18

Hence plot a graph of the function e x over the above range of values.

5 The number of bacteria in a particular culture is given by N 5 1000 3 e 2 t , where the time t is in

minutes. State how long it takes:

a for the number to double

b for the number to increase from 1000 to 4000

c for the number to increase from 1000 to 8000.

d Draw a graph showing the growth of the culture from 1000 to 8000. From your graph find

approximately how long it takes for the number to increase from 1000 to 6000.

6 The electric current through an induction coil is given by I 5 200(1 2 e 2 0.5 t ) mA, where the

time t is in seconds.

a Find the times taken for the current to become 100 mA, 150 mA and 175 mA.

b Draw a graph showing the growth of the current during the first 5 seconds.

c From your graph find the approximate size of the current after 1 second.

SELF-TEST 1 $7000 is invested for 4 years at an interest rate of 5% p.a. What sum, to the nearest cent, does

this grow to:

a if the interest is compounded yearly?

b if the interest is compounded each quarter?

c if the interest is compounded daily?

d if the interest is compounded continuously?

2 A certain bacterial culture grows so that each cell (on average) takes 9.70 minutes for mitosis

(i.e. the ‘doubling time’—the time for one cell to divide into two cells—is 9.70 minutes).

a Calculate the continuous rate of growth k, correct to 3 significant figures.

b If 3.20 mg of bacteria is originally present, how much will be present after 2 hours’ growth?

alL70772_ch16_273-286.indd 285 23/08/12 10:30 AM

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approximately how long it takes for the number to increase from 1000 to 6000. approximately how long it takes for the number to increase from 1000 to 6000.

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The electric current through an induction coil is given by

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The electric current through an induction coil is given by The electric current through an induction coil is given by

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The electric current through an induction coil is given by I

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Find the times taken for the current to become 100 mA, 150 mA and 175 mA.

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Find the times taken for the current to become 100 mA, 150 mA and 175 mA. Find the times taken for the current to become 100 mA, 150 mA and 175 mA.

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Find the times taken for the current to become 100 mA, 150 mA and 175 mA.

Draw a graph showing the growth of the current during the first 5 seconds.

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Draw a graph showing the growth of the current during the first 5 seconds. Draw a graph showing the growth of the current during the first 5 seconds.

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Draw a graph showing the growth of the current during the first 5 seconds.

From your graph find the approximate size of the current after 1 second.

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From your graph find the approximate size of the current after 1 second. From your graph find the approximate size of the current after 1 second.

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From your graph find the approximate size of the current after 1 second.

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$7000 is invested for 4 years at an interest rate of 5% p.a. What sum, to the nearest cent, does Sam

ple

$7000 is invested for 4 years at an interest rate of 5% p.a. What sum, to the nearest cent, does

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Draw a graph showing the growth of the culture from 1000 to 8000. From your graph find Draw a graph showing the growth of the culture from 1000 to 8000. From your graph find only



















Draw a graph showing the growth of the culture from 1000 to 8000. From your graph find

approximately how long it takes for the number to increase from 1000 to 6000. only

approximately how long it takes for the number to increase from 1000 to 6000. approximately how long it takes for the number to increase from 1000 to 6000. only
























approximately how long it takes for the number to increase from 1000 to 6000. only

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3

R

C

E

When the key is closed, the voltage E across

the capacitor C increases exponentially towards

a maximum value according to the equation

E 5 85(1 2 e 2 kt ), where t is in seconds.

a What is the voltage of the source?

b If E rises from 0 to 25 V in the first 730 ms, how

long does it take for E to increase from 0 to 80 V?

4 Draw sketch-graphs of the following curves:

a y 5 5 e 14 x for 0 # x # 0.2

b i 5 40 e 2 100 t mA, where t is in seconds

c Q 5 16(1 2 e 2 500 t ) mC, where t is in seconds

5 A radioactive substance decomposes so that the mass present after t years is given by

M 5 100 e 2 0.1 t g.

a Find, to the nearest year, how long it will take for the mass present to reduce to:

i 50 g ii 25 g iii 12.5 g iv 3.125 g

b Draw a graph showing the decay from 100 g to 3.125 g and from your graph find

approximately the mass present after 10 years.

alL70772_ch16_273-286.indd 286 23/08/12 10:31 AM

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Draw a graph showing the decay from 100 g to 3.125 g and from your graph find

Sample


approximately the mass present after 10 years.

Sample

approximately the mass present after 10 years. onlyA radioactive substance decomposes so that the mass present after

onlyA radioactive substance decomposes so that the mass present after t

onlyt years is given by

only years is given by

Find, to the nearest year, how long it will take for the mass present to reduce to:

only

Find, to the nearest year, how long it will take for the mass present to reduce to:

12.5 g only

12.5 g only

Draw a graph showing the decay from 100 g to 3.125 g and from your graph find only


CHAPTER 23

DETERMINANTS AND MATRICES

There is another method for solving simultaneous linear equations that is applicable to all such

equations and often saves a lot of time. Once learnt, this method is very concise, very simple to use

and provides less opportunity for careless error.

However (there is always a catch, of course), there are a few facts that you will have to learn first.

23.1 DEFINITION AND EVALUATION OF A 2 3 2 DETERMINANT 2a c

b d2 is called a two-by-two determinant. It is an array of numbers having two rows and two

columns enclosed between vertical bars. It is shorthand notation for ad 2 bc.

Note: 2a cb d

2 bc

ad

Learning objectives ■ Evaluate determinants. ■ Solve simultaneous equations using determinants. ■ Multiply matrices. ■ Understand the algebra of matrices. ■ Calculate the inverse of a matrix. ■ Express simultaneous equations in matrix form. ■ Solve simultaneous linear equations using matrices.

Examples

1 22 34 5

2 5 (2 3 5) 2 (4 3 3) 5 22 2 222 324 25

2 5 (22 3 25) 2 (24 3 3) 5 10 1 12 5 22

Exercises 23.1 Evaluate the following 2 3 2 determinants:

1 23 4

2 52 2 22 0

6 12

continued

alL70772_ch23_367-400.indd 367 9/5/12 8:34 PM

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There is another method for solving simultaneous linear equations that is applicable to

Sample

There is another method for solving simultaneous linear equations that is applicable to


Sample



Sample



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23.1 DEFINITION AND EVALUATION OF A 2

Sample

23.1 DEFINITION AND EVALUATION OF A 2 3

Sample

3

two-by-two determinant.

Sample

two-by-two determinant.

Sample

It is an array of numbers having two rows and two

Sample

It is an array of numbers having two rows and two

columns enclosed between vertical bars. It is shorthand notation for Sample

columns enclosed between vertical bars. It is shorthand notation for Sample

Sample

only

There is another method for solving simultaneous linear equations that is applicable to only

There is another method for solving simultaneous linear equations that is applicable to only


23.2 SOLUTION OF SIMULTANEOUS EQUATIONS USING 2 3 2 DETERMINANTS If we have two simultaneous equations, for example: b2x 1 3y 5 4

5x 1 6y 5 7

n is the determinant we obtain from the coefficients on the left-hand sides, in the same order and

arrangement as they appear in the equations.

In this case, ^ 5 22 3

5 62 5 12 2 15 5 23

n x is the same determinant as above, except that the x -coefficients are replaced by the right-hand

numbers of the equations.

In this case, ^x 5 24 3

7 62 5 24 2 21 5 3

n y is the same determinant as n but with the y -coefficients replaced by the right-hand numbers of

the equations.

In this case, ^y 5 22 4

5 72 5 14 2 20 5 26

Examples

1 If b3x 1 5y 5 26x 1 4y 5 7

then 5 23 56 4

2 5 12 2 30

5 218

x 5 22 57 4

2 5 8 2 35

5 227

y 5 23 26 7

2 5 21 2 12

5 9

2 If b 4x 2 3y 5 2522x 2 y 5 6

then 5 2 4 2322 21

2 5 (24) 2 (6)

5 24 2 6

5 210

x 5 225 236 21

2 5 (5) 2 (218)

5 5 1 18

5 23

y 5 2 4 2522 6

2 5 24 2 10

5 14

3 22 6

5 32 4 23 22

5 42

5 2 3 21

24 222 6 26.72 3.91

4.84 5.062

continued

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is the same determinant as above, except that the

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is the same determinant as above, except that the

21

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21 5

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5 3

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3

but with the

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but with the y

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y but with the y but with the

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but with the y but with the -coefficients replaced by the right-hand numbers of

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-coefficients replaced by the right-hand numbers of 2

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22 4

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2 4

5 7

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2Sam

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2 20

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7

only is the determinant we obtain from the coefficients on the left-hand sides, in the same order and

only is the determinant we obtain from the coefficients on the left-hand sides, in the same order and

-coefficients are replaced by the right-hand only

-coefficients are replaced by the right-hand

Determinants and matrices | Chapter 23 369

Exercises 23.2

1 Given that b4x 1 3y 5 1

x 1 2y 5 5

evaluate: a n b n x c n y

2 Given that b m 2 5t 5 3

22m 1 4t 5 21

evaluate: a n b n m c n t

3 Given that

b3.6V 1 2.8e 5 1.8

21.2V 2 5.7e 5 23.3

evaluate: a n b n v c n e

Solution of simultaneous equations Consider any two simultaneous linear equations:

ba1x 1 b1y 5 c1

a2x 1 b2y 5 c2

6 a1b2x 2 b1b2y 5 b2c1

a2b1x 1 b1b2y 5 b1c2

Subtracting: a1b2x 2 a2b1x 5 b2c1 2 b1c2

6 x (a1b2 2 a2b1) 5 b2c1 2 b1c2

6 x 5b2c1 2 b1c2

a1b2 2 a2b1

5

2c1 b1

c2 b222a1 b1

a2 b22

5^x

^

Similarly, it may be shown that y 5^y

^.

3 b2

3 b1

SUMMARY The solution of two simultaneous linear equations:

ba1x 1 b1y 5 c1

a2x 1 b2y 5 c2

is: x 5^x

^, y 5

^y

^

This is known as ‘Cramer’s Rule’. To apply this rule, both equations must be expressed with the constants c 1 and c 2 on the right-hand sides.

alL70772_ch23_367-400.indd 369 05/09/12 10:48 AM

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Example Solve:

b23.5I2 1 16.7I1 2 58.7 5 081.2I1 2 34.2I2 1 13.9 5 0

We first rearrange the equations: b16.7I1 1 23.5I2 5 58.781.2I1 2 34.2I2 5 213.9

5 216.7 23.581.2 234.2

2 5 2571.14 2 1908.2 5 22479.34

I1 5 2 58.7 23.5213.9 234.2

2 5 22007.54 1 326.65 5 21680.89

I2 5 216.7 58.781.2 213.9

2 5 2232.13 2 4766.44 5 24998.57

I1 5^I1

^5

21680.8922479.34

< 0.678, I2 5^I2

^5

24998.5722479.34

< 2.02

Answer: I 1 < 0.678, I 2 < 2.02 Note the advantage of this method. Applying the substitution or elimination method would be very tedious for such equations.

Exercises 23.2 continued Solve the following simultaneous equations correct to 3 significant figures:

4 b7E 1 19V 5 24

13E 2 5V 5 9 5 b13L 2 15W 5 87

17L 2 11W 5 231

6 b2.70x 2 1.40y 5 3.40

3.80x 1 4.60y 5 1.30 7 b29.3I1 2 37.8I2 5 83.7

41.2I1 2 26.4I2 5 216.3

23.3 MATRICES: INTRODUCTION Matrices can be used, among other purposes, to solve simultaneous equations, provided we define

their operations (e.g. addition and multiplication) in special ways. The interest in and use of matrices

has increased greatly since the introduction of computers because their operations are easy to

program on a computer. Manually, for example, it is usually quicker to solve simultaneous equations

using determinants. However, with a computer, matrices are much easier to use, regardless of how

many variables are involved.

Definitions A matrix is a set of numbers (called elements ) arranged in a rectangular pattern (or array ) of rows and

columns. A determinant, as we saw in section 23.1, is this kind of array distinguished by a vertical

bar at each side. To distinguish a matrix, the array is enclosed in parentheses, either round or square.

alL70772_ch23_367-400.indd 370 05/09/12 10:49 AM

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Solve the following simultaneous equations correct to 3 significant figures:

Sample

Solve the following simultaneous equations correct to 3 significant figures: Solve the following simultaneous equations correct to 3 significant figures:

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Solve the following simultaneous equations correct to 3 significant figures:

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5

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55

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3.40

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23.3 MATRICES: INTRODUCTION Sample

23.3 MATRICES: INTRODUCTION

only

only

Note the advantage of this method. Applying the substitution or elimination method would be very only

Note the advantage of this method. Applying the substitution or elimination method would be very only

only

only

only

only

only

only

only

only

only

only

only

only


For example, 22 1

3 42 is a determinant, which has the value 5, but ¢2 1

3 4≤ or B2 1

3 4R is a

matrix, which does not have a value. A matrix does not represent a number.

If a matrix has a rows and b columns, it is said to be an ‘ a 3 b matrix’ or to ‘have an order of

a 3 b ’. A matrix of order ‘ a 3 1’ is called a column matrix. A matrix of order ‘1 3 b ’ is called

a row matrix.

Examples

1 A 5 £ 3 022 4

5 3≥ is a matrix of order 3 3 2.

2 D 5 § 53

212

¥ is a matrix of order 4 3 1, a column matrix.

3 K 5 ¢5 23 2≤ is a matrix of order 1 3 3, a row matrix.

4 P 5 ¢4 220 3

≤ is a 2 3 2 matrix, a square matrix of order 2.

5 T 5 £21 0 32 3 14 3 0

≥ is a 3 3 3 matrix, a square matrix of order 3.

We identify a matrix by a capital letter and its order can be shown under this letter. For

example, K332

is a matrix that we call K and its order is 3 3 2 (i.e. 3 rows and 2 columns).

Exercises 23.3 Below is a set of matrices that are also referred to in following exercises.

A 5 ¢1 0 2

3 2 1≤ B 5 ¢2 1

3 2≤ C 5 £3

1

2

≥

D 5 ¢2 1 3

1 4 2≤ E 5 £1 3 4

2 5 2

4 1 3

≥ F 5 (3 1 2)

G 5 £2 1

1 3

3 2

≥ H 5 ¢1 3

2 1≤ K 5 £3 1 2

0 2 3

1 0 2

≥

1 Using the above set of matrices, state the order of: a D, b G, c C, d A, e F.

2 Which of the matrices in the above set is: a a 2 3 3 matrix, b a 3 3 2 matrix, c a square matrix,

d a row matrix, e a column matrix?

alL70772_ch23_367-400.indd 371 05/09/12 10:49 AM

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matrix of order 2.

Sample

matrix of order 2.

square

Sample

square matrix of order 3.

Sample

matrix of order 3.


Sample


is a matrix that we call

Sample

is a matrix that we call K

Sample

K and its order is 3

Sample

and its order is 3 K and its order is 3 K

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K and its order is 3 K

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Below is a set of matrices that are also referred to in following exercises. Sample

Below is a set of matrices that are also referred to in following exercises. Below is a set of matrices that are also referred to in following exercises. Sample























Below is a set of matrices that are also referred to in following exercises. Sample

Sample

only


23.4 SOME DEFINITIONS AND LAWS Equal matrices Two matrices are said to be equal if, and only if, they are identical in every respect—that is, the

elements of each are the same and in the same positions.

Example

If ¢a bc d

≤ 5 ¢2 41 3

≤ , then a 5 2, b 5 4, c 5 1, d 5 3.

The sum or difference of two matrices These are found by adding or subtracting the corresponding elements of each matrix.

Example ¢a b cd e f

≤ 1 ¢g h ij k l

≤ 5 ¢a 1 g b 1 h c 1 id 1 j e 1 k f 1 l

≤¢5 3 12 4 0

≤ 2 ¢3 3 05 1 2

≤ 5 ¢ 2 0 123 3 22

≤

Note: Two matrices can be added or subtracted only when they have the same order (i.e. they must have the same number of rows and the same number of columns; they must have the ‘same shape’). The resulting sum or difference will also have the same order.

From the definition, it can be seen that A 1 B 5 B 1 A (the commutative law for addition).

The zero matrix

For matrix A the zero matrix, O, is defined to be the matrix such that A 1 O 5 O 1 A 5 A (the law of addition of zero). The letter O is used to denote a zero matrix.

Example

The zero matrix of ¢a b cd e f

≤ is ¢0 0 00 0 0

≤ . The zero matrix of ¢2 230 5

≤ is ¢0 00 0

≤ .

Note: The matrix O is not the number zero but is the matrix of the same order as A, which has the number 0 for each of its elements.

alL70772_ch23_367-400.indd 372 05/09/12 10:49 AM

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l≤

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2

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22

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2

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≤

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≤

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Two matrices can be added or subtracted only when they have the same order (i.e. they must

Sample

Two matrices can be added or subtracted only when they have the same order (i.e. they must

have the same number of rows and the same number of columns; they must have the ‘same shape’).

Sample

have the same number of rows and the same number of columns; they must have the ‘same shape’). The resulting sum or difference will also have the same order.

Sample

The resulting sum or difference will also have the same order.

Sample

From the definition, it can be seen that Sample

From the definition, it can be seen that

The zero matrix Sam

ple

The zero matrix

only These are found by adding or subtracting the corresponding elements of each matrix.

only These are found by adding or subtracting the corresponding elements of each matrix.

only

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Multiplication by a constant By definition, a matrix is multiplied by a constant by multiplying every element of the matrix by

that constant.

Example

3 3 ¢2 21 00 3 22

≤ 5 ¢6 23 00 9 26

≤

Exercises 23.4 1 Solve the following matrix equations:

a ¢x

y≤ 5 ¢7

3≤ b ¢x 1 2

y 2 3≤ 5 ¢5

6≤

c ¢ x 2 1

2y 1 3≤ 5 ¢3 2 x

y 1 9≤ d ¢x 1 y

x 2 y≤ 5 ¢3

5≤

2 Write the single matrix equation

£3x 1 2y 2 5z

4x 2 3y 1 2z

5x 1 5y 2 3z

≥ 5 £8

7

9

≥

as three separate simultaneous equations.

3 Using the set of matrices (A, B, C, . . ., K) in Exercises 23.3:

a state which pairs of those matrices can be added or subtracted

b write down the matrix K 1 E

c state the zero matrix of D

d state the zero matrix of E

e write down the matrix 3 3 A

f write down the matrix 2 E 1 3 K

23.5 MULTIPLICATION OF MATRICES Since a matrix is simply an array (arrangement) of numbers in a rectangular pattern and does not

have a value, we can define the product of two matrices in any way we choose.

For the purposes of explanation, the rows and columns of a matrix will be designated as shown

in the matrix below: the rows being called R 1 , R 2 , R 3 , . . . and the columns C 1 , C 2 , C 3 , . . . .

Any particular element of a matrix can be identified by stating its row and its column.

C1 C2 C3

T T T

For example, in the matrix R1 SR2 S

¢3 6 7

2 5 4≤ , R1C3 5 7, R1C1 5 3 and R2C2 5 5 .

alL70772_ch23_367-400.indd 373 05/09/12 10:49 AM

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By definition, when two matrices are multiplied, the product is another matrix and regardless of

how many rows and columns the matrices may possess, when the elements in R n of the first matrix

are multiplied in succession by the elements of C m of the second matrix and these products are

added, this gives element R n C m of the product matrix.

When put into words this definition seems very complicated but some illustrations and some

practice should enable you to gain facility with this process.

Ignoring all the other rows and columns that may be present:

C2

T

C2

T

1 R3 S § 3 3 3

3 3 3

5 2 7

3 3 3

¥ 3 § 3 4 3

3 0 3

3 1 3

¥ 5 R3 S § 3 3 3

3 3 3

3 27 3

3 3 3

¥

In the product matrix, element R 3 C 2 5 (5 3 4) 1 (2 3 0) 1 (7 3 1) 5 27

2 ¢1 2

3 4≤ 3 ¢5 7 9

6 8 10≤ 5 ¢(1 3 5) 1 (2 3 6) (1 3 7) 1 (2 3 8) (1 3 9) 1 (2 3 10)

(3 3 5) 3 (4 3 6) (3 3 7) 1 (4 3 8) (3 3 9) 1 (4 3 10)≤

5 ¢ 5 1 12 7 1 16 9 1 20

15 1 24 21 1 32 27 1 40≤

5 ¢17 23 29

39 53 67≤

You are advised to practise this process until it becomes quite familiar to you. Below are some

exercises to enable you to practise the multiplication of two matrices. You will quickly discover

the benefit of using fingers or a pen to obscure the rows and columns not being used to obtain a

particular element of the product matrix.

Example

¢2 34 5

≤ 3 ¢4 21 3

≤ 5 ¢ (2 3 4) 1 (3 3 1) (2 3 2) 1 (3 3 3)

(4 3 4) 1 (5 3 1) (4 3 2) 1 (5 3 3)≤

5 ¢ 8 1 3 4 1 916 1 5 8 1 15

≤ 5 ¢11 13

21 23≤

When the elements are small you should be able to obtain the product matrix without needing to

write down the intermediate steps.

e.g. ¢3 1

2 4≤ 3 ¢2 0

1 3≤ 5 ¢7 3

8 12≤

alL70772_ch23_367-400.indd 374 05/09/12 10:49 AM

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Exercises 23.5 1 a You are given the following matrices:

A ¢3 1

2 4≤ , B ¢5 1

3 2≤ , C ¢1 2

4 3≤ , D ¢5 3

2 1≤ , E ¢2 0

0 3≤

Write down the following matrices:

i A 3 B ii A 3 C iii A 3 D iv A 3 E

v B 3 C vi B 3 D vii B 3 E viii C 3 D

ix C 3 E x D 3 E

b You are given the following matrices:

P ¢21 3

22 4≤ , Q ¢3 22

6 24≤ , R ¢4 23

2 21≤ , S ¢22 0

23 0≤ , T ¢24 2

26 3≤

Write down the following matrices:

i P 3 Q ii P 3 R iii P 3 S iv P 3 T

v Q 3 R vi Q 3 S vii Q 3 T viii R 3 S

ix R 3 T x S 3 T

2 As always in algebra, if there is no operation sign between two terms, multiplication is to be

assumed. AK means A 3 K, i.e. matrix A 3 matrix K.

Find the product of the matrices in each case below:

a £1 3 0

0 2 1

2 0 3

≥ £1 2 3

2 0 2

0 3 1

≥ b £21 2 21

0 0 3

22 0 1

≥ £22 21 0

2 23 0

1 0 21

≥

c £2 1 0

1 0 3

3 1 1

≥ £0 1 2

1 1 0

2 3 1

≥ d ¢ 1 0

21 2≤ ¢3 22

1 5≤

e ¢1 0

3 2≤ ¢5 4 8

6 7 9≤ f ¢0 1 1

1 3 23≤ £2 1 21

4 3 0

0 21 5

≥

g ¢1 2

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0 2 21≤ £1 5

4 2

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≥

i (0 3 2) £ 1

21

4

≥ j £ 2

21

3

≥ (22 3 21)

continued

alL70772_ch23_367-400.indd 375 05/09/12 10:49 AM

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23.6 COMPATIBILITY By now it has probably become clear to you that multiplication of two matrices is only possible

when the number of columns in the first matrix equals the number of rows in the second matrix. i.e.

the product Am3n

3 Bp3q

5 C exists only when n 5 p. When, and only when, this is so, the matrices

are said to be ‘compatible’ for this multiplication. If n ? p, the matrices cannot be multiplied and are

said to be ‘incompatible’ for this operation.

k ¢2 1

4 3≤ ¢5

6≤ l (7 22 5) £ 6

22

8

≥

m £ 2 0

21 3

22 4

≥ ¢23

0≤ n (4 5 1) £ 0

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3

≥

o ¢22 21

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≥ £24 0

3 22

0 1

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3 If A 5 £3 21 2

0 2 1

1 3 0

≥ and B 5 ¢21 2

2 0≤ , find: a A2, b B3.

continued

Examples 1 The product M 3 N does exist

and has order 2 3 5.

2 The product P 3 Q does not exist.

compatible

3 5

N

order of product

2

M

3

incompatible

2

Q

2

P

2 3

compatible

33

T

4

R

3

order of product

3 The product R 3 T does exist and has order 4 3 3.

alL70772_ch23_367-400.indd 376 05/09/12 10:49 AM

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It is quite common for a product A 3 B to exist but for the product B 3 A not to exist.

For example:

V 3 W exists (and has order 2 3 3), but

2 3 3 3 3 3

W 3 V does not exist.

3 3 3 2 3 3

It is easy to show that A 3 B and B 3 A both exist only for Am3n

3 Bn3m

:

A 3 B B 3 A

e.g. and both exist.

2 3 3 3 3 2 3 3 2 2 3 3

SUMMARY Facts about the product of two matrices:

■ A 3 B may not exist.

■ Even if A 3 B does exist, it is possible that B 3 A does not exist.

■ Even when both products exist, in general, A 3 B ? B 3 A.

■ It is possible that, A 3 B 5 0 even though neither A nor B is the zero matrix O.

Example ¢3 26 4

≤ 3 ¢ 4 2226 3

≤ 5 ¢0 00 0

≤ 5 0

You can verify this after studying the next section.

Exercises 23.6 1. Using the set of matrices in Exercises 23.3, state whether the given product exists in each case

(answering ‘yes’ or ‘no’) and, if it does exist, state its order.

a AB b BA c AC d DA e FE f EF

g CF h AE i BD j CD k DC l CE

23.7 THE IDENTITY MATRIX, I Exercises 23.7 1 Write down the product matrix:

a ¢a b

c d≤ ¢1 0

0 1≤ b ¢1 0

0 1≤ ¢a b

c d≤

continued

alL70772_ch23_367-400.indd 377 05/09/12 10:49 AM

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Using the set of matrices in Exercises 23.3, state whether the given product exists in each case

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Using the set of matrices in Exercises 23.3, state whether the given product exists in each case Using the set of matrices in Exercises 23.3, state whether the given product exists in each case

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The principal diagonal of a square matrix is the diagonal that runs from the top left-hand corner to

the bottom right-hand corner. The square matrix, which has the number 1 for each element on the

principal diagonal and all other elements zero, plays a very special role in the theory of matrices. ¢1 0

0 1≤ is called the identity matrix of order 2, and is specifi ed as I 2 . £1 0 0

0 1 0

0 0 1

≥ is called the identity matrix of order 3, and is specifi ed as I 3 .

For any square matrix A n (i.e. of order n 3 n ), A n 3 I n 5 I n 3 A n .

I n plays the same role in matrix theory as unity does in arithmetic (e.g. 7 3 1 5 1 3 7 5 7), and

so it is called the unit matrix, U n or the identity matrix, I n . We use the latter name and symbol in

this book.

We use capital letters to identify matrices, but the capital letter I is reserved for the identity

matrix and the capital letter O is reserved for the zero matrix.

2 Write down the product matrix:

a £a b c

d e f

g h i

≥ £1 0 0

0 1 0

0 0 1

≥ b £1 0 0

0 1 0

0 0 1

≥ £a b c

d e f

g h i

≥

continued

Note: For a matrix An3m

which is not square, An3m

3 Im 5 An3m

(but Im 3 An3m

does not exist) and

In 3 An3m

5 An3m

(but An3m

3 In does not exist).

Remember: I stands for the I dentity matrix, not the numeral 1.

Exercises 23.7 continued

3 For the following, write down the product matrix if it exists. If it does not exist, write

‘incompatible’.

a ¢1 2 3

4 5 6≤ 3 £1 0 0

0 1 0

0 0 1

≥ b £1 0 0

0 1 0

0 0 1

≥ 3 ¢2 1 3

1 3 2≤

c ¢1 0

0 1≤ ¢1 2 3

4 5 6≤ d ¢1 2 3

4 5 6≤ ¢1 0

0 1≤

4 a If ¢17 213 19

34 28 215≤ 3 A 5 ¢17 213 19

34 28 215≤ , write down the matrix A.

Remember: an identity matrix is always square.

Therefore, non-square matrices do not have an identity matrix.

alL70772_ch23_367-400.indd 378 05/09/12 10:49 AM

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We use capital letters to identify matrices, but the capital letter I is reserved for the

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23.8 THE INVERSE MATRIX, A 2 1 The numbers 13 and 113 are said to be multiplicative inverses of one another because in multiplication

one undoes what the other does.

For example, 957 3 13 3113 5 957, "7 3

113 3 13 5 "7.

This occurs because 13 3113 5 1 and 113 3 13 5 1.

Now we will see how this applies to matrices.

b If B 3 ¢17 213 19

34 28 215≤ 5 ¢17 213 19

34 28 215≤ , write down the matrix B.

c Does ¢17 213 19

34 28 215≤ have an identity matrix? State the reason for your answer.

5 For each of the following matrices, state whether an identity matrix exists (answering ‘yes’ or ‘no’)

and, if it does exist, write it down.

a ¢23 2

7 5≤ b £22 8

7 3

1 0

≥

c ¢3 24 2

6 1 25≤ d £ 3 22 5

4 1 6

27 9 8

≥

6 If M 5 ¢0 21

1 0≤ , show that: a M 2 5 2 I , b M 3 5 2 M.

Definition: Two matrices, A and B, are said to be inverses of one another (i.e. A 5 B 2 1 and B 5 A 2 1 ) if AB 5 BA 5 I.

Exercises 23.8

1 If A 5 ¢5 23

2 21≤ and B 5 ¢21 3

22 5≤ , write down: a the matrix AB , b the matrix BA.

Since both products in the exercise above are the identity matrix, you can probably guess that A

and B are said to be inverses of each other.

The inverse of matrix M is written as M 2 1 , so you have proved for the above matrices A and B

that AB 5 BA 5 I, that is, that B 5 A 2 1 and A 5 B 2 1 .

alL70772_ch23_367-400.indd 379 05/09/12 10:49 AM

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are said to be multiplicative

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Most square matrices have an inverse, but not all of them. (Actually, it can be shown that all

square matrices have an inverse except those for which the determinant | A| 5 0.)

A matrix that has an inverse is said to be invertible.


2 Given that C 5 £1 1 1

1 1 2

1 2 1

≥ and D 5 £ 3 21 21

21 0 1

21 1 0

≥:

a write down the matrix CD

b write down the matrix DC

c what have we proved about the matrices C and D?

3 Given that matrix Ap3q

has an inverse Br3s

:

a what is the order of matrix AB?

b what is the order of matrix BA?

c Since these matrices are inverses of each other, by definition AB 5 BA 5 In3n

. Therefore, the

orders of AB and BA are both n 3 n. What can you deduce about the values of p, q, r, s and n?

d Hence, if matrices A and B are inverses of each other, what can you deduce about the shapes

of the matrices A and B?

SUMMARY ■ If A 3 B 5 B 3 A 5 I, then A and B are inverses of each other (i.e. A 5 B 2 1 and B 5 A 2 1 ).

■ A non-square matrix cannot have an inverse.

■ Most, but not all, square matrices have an inverse (i.e. they are invertible ) .

Note: You are not required to be able to find the inverse A 2 1 of a given matrix A, but you should be able to determine whether or not two given matrices A and B are inverses of each other by testing whether AB 5 BA 5 I.


4 If A 5 £2 1 1

1 1 1

2 2 1

≥ B 5 £1 2 22

2 5 24

3 7 25

≥ C 5 £ 1 21 0

21 0 1

0 2 21

≥

D 5 £1 2 0

0 1 2

1 0 2

≥ E 5 £ 3 24 2

22 1 0

21 21 1

≥:

alL70772_ch23_367-400.indd 380 05/09/12 10:49 AM

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square matrices have an inverse, but not

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square matrices have an inverse, but not all

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all of them. (Actually, it can be shown that all

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of them. (Actually, it can be shown that all

except those for which the determinant

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except those for which the determinant

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invertible.

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are inverses of each other (i.e.

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A non-square matrix cannot have an inverse.

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A non-square matrix cannot have an inverse.

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onlyAA

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only are inverses of each other, what can you deduce about the


23.9 THE ALGEBRA OF MATRICES As a result of the definitions of the operations of matrices, the laws for the algebra of matrices are

mostly the same as the laws for the algebra of real numbers.

Name of law Real numbers Matrices

Commutative law for addition x 1 y 5 y 1 x A 1 B 5 B 1 A

Associative law for addition (x 1 y) 1 z 5 x 1 (y 1 z) (A 1 B) 1 C 5 A 1 (B 1 C)

Identity law for addition x 1 0 5 0 1 x 5 x A 1 O 5 O 1 A 5 A

(0 is the identity element for addition)

(O is the identity matrix for addition, the zero matrix of A)

Identity law for multiplication x 3 1 5 1 3 x 5 x A 3 I 5 I 3 A 5 A

(1 is the identity element for multiplication)

(I is the identity matrix for multiplication for matrix A)

Law of multiplicative inverse x 3 x21 5 x21 3 x 5 1 A 3 A21 5 A21 3 A 5 I

(x21 is the multiplicative inverse of x)

(A21 is the multiplicative inverse of A)

Distributive law for multiplication C(x 1 y) 5 Cx 1 Cy k(A 1 B) 5 kA 1 kB

(x 1 y)C 5 xC 1 yC (A 1 B)k 5 Ak 1 Bk

x(y 1 z) 5 xy 1 xz A(B 1 C) 5 AB 1 AC

(y 1 z)x 5 yx 1 zx (B 1 C)A 5 BA 1 CA

a write down the products:

i AB ii AC iii AD iv AE

v BC vi BD vii BE

b hence, write down: (i) matrix A 2 1 (ii) matrix B 2 1

5 Given that J 5 £1 2 21

0 3 22

0 0 21

≥ and K 5 £3 22 1

0 1 22

0 0 23

≥:

a write down the matrix JK

b write down the matrix J 2 1

6 Given that P 5 £1 2 22

1 3 21

3 2 0

≥ and Q 5 £ 2 24 4

23 6 21

27 4 1

≥:

a write down the matrix PQ

b write down the matrix Q 2 1

alL70772_ch23_367-400.indd 381 05/09/12 10:49 AM

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As a result of the definitions of the operations of matrices, the laws for the algebra of matrices are

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Remember:

■ O, the zero matrix for matrix A, has been defi ned as the matrix with the same order as A but having all its elements zeros.

■ I, the identity matrix for matrix A, has been defi ned for square matrices only, being the matrix having the same order as A, with all the elements on the principal diagonal being 1 and all the other elements being zero.

■ A 2 1 , the inverse of matrix A, has been defi ned for square matrices only, being the matrix such that AA 2 1 5 A 2 1 A 5 I. The only matrices that have an inverse are square matrices whose determinant ? 0.

Hence, most algebraic operations with matrices are already quite familiar to us.

Examples

1 If 3A 1 4B 5 5C

then 3A 5 5C 2 4B [ A 5

13(5C 2 4B)

2 ( A 1 B )( C 1 D ) 5 AC 1 AD 1 BC 1 BD

3 (A 1 B)(A 1 I) 5 A2 1 AI 1 BA 1 BI

5 A2 1 A 1 BA 1 B

4 A(A21 1 I) 5 AA21 1 AI

5 I 1 A, or A 1 I 5 A ( A 2 1 B ) 5 ( AA 2 1 ) B 5 IB 5 B

6 AB 1 A 5 AB 1 AI

5 A(B 1 I), not A(B 1 1) because we cannot add a real number to a matrix

7 A21(A 1 I) 5 A21A 1 A21I

5 I 1 A21, or A21 1 I

However, there is a difference between the algebra for matrices and the algebra for real numbers

when we are multiplying because, as we have already noted, in general:

A 3 B 2 B 3 A.

(Don’t forget that this statement does not mean that they can never be equal but that we cannot

assume they are equal, because usually they are not equal.)

Hence, care must be taken to maintain the correct order of matrices when multiplying.

For example, A ( B 1 C ) ? ( B 1 C ) A and ABA 2 1 ? A 2 1 AB (which would equal B ).

The only occasion when multiplying matrices is commutative is when they are inverses of one

another ( AA 2 1 5 A 2 1 A [ 5 I ]) or when one of them is the identity matrix ( AI 5 IA [ 5 A ]).

Example Make K the subject of the equation AK 5 B.

We proceed thus: AK 5 B

6 A21(AK) 5 A21B (not BA21)

6 (A21A)K 5 A21B

6 K 5 A21B

alL70772_ch23_367-400.indd 382 05/09/12 10:49 AM

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6 AF 1 2IF 5 B

(A 1 2I)F 5 B

6 (A 1 2I)21(A 1 2I)F 5 (A 1 2I)21B

6 F 5 (A 1 2I)21B

Exercises 23.9 1 Simplify, removing all brackets:

a A ( A 2 1 B ) b A ( A 2 1 I )

c A ( BA 2 1 ) 1 A ( A 2 1 B ) d AB ( A 2 1B 1 B 2 1 A )

e I 2 f I 2 1 I 3

g ( A 1 I ) 2 h ( A 1 I )( B 1 A )

i A ( I 2 A 2 1 ) 1 I 2 j A ( A 2 1 1 I ) 2 B 2 1 B 2 IA

2 Solve the following matrix equation for X:

a 2 X 2 A 5 B b C 2 3 X 5 D

c 3( A 2 2 X ) 5 2(3 A 2 X ) d X 1 A

35

X 2 3B

2

3 Solve the following matrix equations for X:

a AX 5 B b XA 5 B

c 2 A 1 AX 5 B d AX 1 I 5 B

e X 2 1 5 A f 3 X 2 AX 5 B

g AX 1 X 5 B h AX 1 B 5 C 2 X

i X 1 XA 5 B j 2 X 1 B 5 C 2 2 XA

4 If AB 5 AC:

a does it follow that B 5 C (‘yes’ or ‘no’)?

b and B ? C, what is B equal to?

5 If AB 5 CA:

a does it follow that B 5 C (‘yes’ or ‘no’)?

b and B ? C, what is B equal to?

alL70772_ch23_367-400.indd 383 05/09/12 10:49 AM

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Note:

■ The matrix, A 3 B 3 C 5 ( A 3 B ) 3 C 5 A 3 ( B 3 C ), the associative law, but we must not change the order of the matrices when they are being multiplied.

■ The matrices ABC, ACB, BCA, BAC, CAB and CBA are probably all different matrices. ■ However, consideration will show you that if k is a constant (i.e. a real number), then

k 3 ( A 3 B ) 5 ( k 3 A ) 3 B 5 A 3 ( k 3 B ). Which matrix we multiply by k, either A 3 B or A or B, makes no difference to the fi nal result. Although we must not change the positions of the matrices, we may change the position of a constant, k.

■ Note also that k 3 A 5 k 3 AI 5 A 3 k l.


6 Solve the following matrix equations for X:

a 3 X 1 2 XA 5 C 2 B b A 1 2 X 5 B 1 3 XC

23.10 EXPRESSING SIMULTANEOUS EQUATIONS IN MATRIX FORM Exercises 23.10 1 Find the following products and state their order:

a ¢2 3

5 4≤ ¢x

y≤ b £3 5 22

1 4 3

2 23 4

≥ £x

y

z

≥

2 Express as the product of two matrices:

a ¢3x 1 4y

2x 2 5y≤ b £2x 2 3y 1 4z

x 1 2y 2 5z

3x 2 y 1 2z

≥ c £3a 2 5b 1 2c

2a 1 4b

5a 2 7c

≥

3 Solve for x and y:

a ¢2x

3y≤ 5 ¢ 8

29≤ b ¢13 1

12 1≤ ¢x

y≤ 5 ¢29

27≤

4 Express as a system of simultaneous equations:

a ¢2x 1 3y

5x 2 2y≤ 5 ¢3

4≤ b ¢3 5

2 24≤ ¢x

y≤ 5 ¢6

7≤

c £3x 2 2y 1 4z

2x 1 3y 2 5z

x 2 4y 1 2z

≥ 5 £6

7

8

≥ d £1 2 3

0 1 2

1 22 0

≥ £x

y

z

≥ 5 £4

5

6

≥

5 Express each of the following systems of simultaneous equations as a single matrix equation:

a c5a 1 7b 2 3c 5 17

7a 2 2b 2 8c 5 13

3a 1 5b 1 5c 5 19

b c4x 2 5y 1 6z 5 9

7x 1 3y 5 5

3x 2 8z 5 7

alL70772_ch23_367-400.indd 384 05/09/12 10:49 AM

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23.11 SOLVING SIMULTANEOUS LINEAR EQUATIONS USING MATRICES If a set of simultaneous equations is represented in matrix form

C 3 U 5 K

where:

C is the matrix formed by the coeffi cients

U is the matrix formed by the unknowns

K is the matrix formed by the constants

then the solution to the matrix of unknown constants U can be solved by multiplying both sides of

the matrix equation by the inverse of C, the matrix of coefficients.

C21 3 C 3 U 5 C21 3 K

We know that:

C21 3 C 5 I ( the identity matrix)

Therefore,

I 3 U 5 C21 3 K

We also know that:

I 3 U 5 U

Therefore the solution to U, the matrix of unknown constants, becomes:

U 5 C21 3 K

This is a simple but powerful equation. In order to find the solution to a matrix of unknowns we

only need to determine the inverse of the matrix of coefficients.

Solving two equations with two unknowns For the general equation ba1x 1 b1y 5 c1

a2x 1 b2y 5 c2

represented in matrix equation form by ¢a1 b1

a2 b2≤ ¢x

y≤ 5 ¢c1

c2≤ ,

C 5 ¢a1 b1

a2 b2≤ , U 5 ¢x

y≤ , K 5 ¢c1

c2≤

C 3 U 5 K or U 5 C21 3 K

Example Solve the simultaneous equations b5x 2 2y 5 6

3x 2 y 5 5

given that the inverse of ¢5 223 21

≤ is ¢21 223 5

≤ .

continued

alL70772_ch23_367-400.indd 385 05/09/12 10:49 AM

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Steps 1 Express the equations as a single matrix equation:

¢5 223 21

≤ ¢xy≤ 5 ¢6

5≤

2 Multiply both sides by the inverse matrix:

¢21 223 5

≤ ¢5 223 21

≤ ¢xy≤ 5 ¢21 2

23 5≤ ¢6

5≤

3 Multiply out the matrices on each side—we know the product on the left-hand side because

A21A 5 I 5 ¢1 00 1

≤ .

¢1 00 1

≤ ¢xy≤ 5 ¢4

7≤

6 ¢xy≤ 5 ¢4

7≤

6 x 5 4, y 5 7

continued

Warning:

The error most often made is in step 3. Remember that ¢21 223 5

≤ ¢65≤ is not the same as

¢65≤ ¢21 2

23 5≤ because for matrices A 3 B ? B 3 A.

Finding the inverse of a 2 3 2 matrix In order to find the solution to the matrix of unknowns U we need to find the inverse of the matrix

of coefficients C 2 1 .

To solve for C 2 1 we manipulate the matrix of coefficients C using the following steps:

1 Interchange the elements on the principal diagonal.

2 Reverse the signs of the elements on the secondary diagonal.

3 Divide by the determinant of the original matrix.

For C 5 ¢a1 b1

a2 b2≤ , C21 5

10C 0 ¢ b2 2b1

2a2 a1≤

Example Solve the following set of linear equations:

24x 1 3y 5 2

22x 1 y 5 4 Representing in matrix form: ¢24 3

22 1≤ 3 ¢x

y≤ 5 ¢2

4≤

C 3 U 5 K

alL70772_ch23_367-400.indd 386 05/09/12 10:49 AM

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The error most often made is in step 3. Remember that

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The error most often made is in step 3. Remember that ¢

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Finding the inverse of a 2

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Finding the inverse of a 2 3

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3 2 matrix

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2 matrix

In order to find the solution to the matrix of unknowns

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In order to find the solution to the matrix of unknowns

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1

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1 we manipulate the matrix of coefficients

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we manipulate the matrix of coefficients

Interchange the elements on the principal diagonal. Sample

Interchange the elements on the principal diagonal.

Reverse the signs of the elements on the secondary diagonal. Sample

Reverse the signs of the elements on the secondary diagonal. Sample

Divide by the determinant of the original matrix. Sample

Divide by the determinant of the original matrix. ≤Sam

ple

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Manipulate the matrix of the coefficients C:

C 5 ¢24 322 1

≤

Interchange the elements on the principal diagonal and reverse the signs of the elements on the secondary diagonal.

C21 510C 0 ¢1 23

2 24≤

Calculate the determinant of the matrix of coefficients.

0C 0 5 224 322 1

2 5 Q24R 2 Q26R 5 2

Divide by the determinant of the matrix of coefficients.

C21 512

¢1 232 24

≤

Now that we have determined C 2 1 we can continue and solve for the matrix of unknowns U. Multiply both sides of the matrix equation by C 2 1 :

C21 3 C 3 U 5 C21 3 K

U 5 C21 3 K

¢xy≤ 5

12

¢1 232 24

≤ ¢24≤

¢xy≤ 5 £ 1

2 3 (2 2 12)12 3 (4 2 16)

≥ 5 ¢2526≤

6 x 5 25, y 5 26

Note: If | A| 5 0, the matrix A has no inverse (because division by zero is not defined). Therefore, the matrix is not invertible.

Solving three equations with three unknowns The general equations are: ca1x 1 b1y 1 c1z 5 d1

a2x 1 b2y 1 c2z 5 d2

a3x 1 b3y 1 c3z 5 d3

Representing in matrix form: £a1 b1 c1

a2 b2 c2

a3 b3 c3

≥ £x

y

z

≥ 5 £d1

d2

de

≥C 5 £a1 b1 c1

a2 b2 c2

a3 b3 c3

≥, U 5 £x

y

z

≥, K 5 £d1

d2

de

≥

C 3 U 5 K, where U is the matrix of unknowns, given by U 5 C 2 1 3 K

alL70772_ch23_367-400.indd 387 05/09/12 10:49 AM

Sample

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has no inverse (because division by zero is not defined). Therefore, the matrix

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has no inverse (because division by zero is not defined). Therefore, the matrix

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Solving three equations with three unknowns Sample

Solving three equations with three unknowns The general equations are: Sam

ple

The general equations are:

only we can continue and solve for the matrix of unknowns

only we can continue and solve for the matrix of unknowns U

onlyU.

only.


Example

If M 5 £ 1 2 213 5 21

22 21 22≥ and N 5

12

£ 11 25 2328 4 227 3 1

≥ :

a find the matrix MN b write down the matrix M 2 1 c express the given system linear equations as a single matrix equation and use the result of b above to

solve these simultaneous equations: c x 1 2y 2 z 5 213x 1 5y 2 z 5 2

22x 2 y 2 2z 5 29

Solutions

a MN 512

£2 0 00 2 00 0 2

≥ 5 £1 0 00 1 00 0 1

≥

b M21 512

£ 11 25 2328 4 227 3 1

≥

c £ 1 2 213 5 21

22 21 22≥ £x

yz≥ 5 £21

229≥

M 3 £xyz≥ 5 £21

229≥

M2 1M 3 £xyz≥ 5

12

£ 11 25 2328 4 227 3 1

≥ £212

29≥

512

£ 62224≥

6 £xyz≥ 5 £ 3

212≥

6 x 5 3, y 5 21, z 5 2

alL70772_ch23_367-400.indd 388 05/09/12 10:49 AM

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Finding the inverse of a 3 3 3 matrix As with a 2 3 2 matrix, the solution to U, the matrix of unknowns, first requires us to be able to

determine C 2 1 . Again, to find C 2 1 we manipulate the matrix C, but in the case of a 3 3 3 matrix

the manipulation is more complex. The solution however is highly methodical and lends itself to the

use of spreadsheets or computer programs. The inverse of a 3 3 3 matrix of coefficients, C21, 5 the

matrix of minors 4 the determinant of the matrix of coefficients.

C21 510C 0 3 ¶ 2b2

b3

c2

c32 2 2b1

b3

c1

c32 2b1

b2

c1

c22

2 2a2

a3

c2

c32 2a1

a3

c1

c32 2 2a1

a2

c1

c222a2

a3

b2

b32 2 2a1

a3

b1

b32 2a1

a2

b1

b22 ¶

where 0C 0 is the determinant of the matrix of coefficients

and 0C 0 5 a1 2b2 c2

b3 c32 2 b1 2a2 c2

a3 c32 1 c1 2a2 b2

a3 b32

For computer applications such as spreadsheets and programs the above can be used directly and

the calculations are straightforward. For hand calculations however it becomes difficult to remember

the position of the determinants in the matrix and the order of the elements in the determinants, so

a stepwise approach for hand calculations is the best approach as follows.

1 Calculate the matrix of minors. To develop the matrix of minors, each element in the matrix is

replaced with the determinant of corresponding matrix elements. The corresponding elements

are the elements that are not in the same row or the same column as the element being replaced.

¶

2b2

b3

c2

c32 2a2

a3

c2

c32 2a2

a3

b2

b322b1

b3

c1

c32 2a1

a3

c1

c32 2a1

a3

b1

b322b1

b2

c1

c22 2a1

a2

c1

c22 2a1

a2

b1

b22 ¶

£a1 b1 c1

a2 b2 c2

a3 b3 c3

≥ £

a1 b1 c1

a2 b2 c2

a3 b3 c3

≥Element replaced with the

determinant of corresponding

elements

Element a 3 is replaced with the determinant of elements b 1 , c 1 , b 2 , c 2 .

Element c 2 is replaced with the determinant of elements a 1 , b 1 , a 3 , b 3 .

2 The sign of each element in the matrix of minors is modified according to the following

matrix of coefficients.

The matrix of minors

alL70772_ch23_367-400.indd 389 05/09/12 10:49 AM

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Calculate the matrix of minors. To develop the matrix of minors, each element in the matrix is

Sample

Calculate the matrix of minors. To develop the matrix of minors, each element in the matrix is


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are the elements that are not in the same row or the same column as the element being replaced.

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are the elements that are not in the same row or the same column as the element being replaced. 2 2

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a

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For computer applications such as spreadsheets and programs the above can be used directly and only


the calculations are straightforward. For hand calculations however it becomes difficult to remember only



C1 C2 C3

R1

R2

R3

£1 2 1

2 1 2

1 2 1

≥

This means that the element in R 1 C 1 remains unchanged, i.e. it is multiplied by 1 1. The element

in R 1 C 2 changes sign, i.e. it is multiplied by 2 1.

¶ 2b2

b3

c2

c32 2 2a2

a3

c2

c32 2a2

a3

b2

b32

2 2b1

b3

c1

c32 2a1

a3

c1

c32 2 2a1

a3

b1

b322b1

b2

c1

c22 2 2a1

a2

c1

c22 2a1

a2

b1

b22 ¶

3 The matrix is transposed. This means that the elements in the columns of the matrix become the

elements in the rows and vice versa.

¶ 2b2

b3

c2

c32 2 2b1

b3

c1

c32 2b1

b2

c1

c22

2 2a2

a3

c2

c32 2a1

a3

c1

c32 2 2a1

a2

c1

c222a2

a3

b2

b32 2 2a1

a3

b1

b32 2a1

a2

b1

b22 ¶

4 The fourth and final step is to divide this manipulated matrix by the determinant of the original

matrix. This provides us with the inverse of the matrix.

C21 510C 0 ¶ 2b2

b3

c2

c32 2 2b1

b3

c1

c32 2b1

b2

c1

c22

2 2a2

a3

c2

c32 2a1

a3

c1

c32 2 2a1

a2

c1

c222a2

a3

b2

b32 2 2a1

a3

b1

b32 2a1

a2

b1

b22 ¶

where:

0C 0 5 a1 2b2 c2

b3 c32 2 b1 2a2 c2

a3 c32 1 c1 2a2 b2

a3 b32

Example Solving the following set of linear equations: c x 2 2y 1 3z 5 4

3x 1 y 2 2z 5 274x 2 4y 1 3z 5 23

Representing in matrix form £1 22 33 1 224 24 3

≥ £xyz≥ 5 £ 4

2723≥

alL70772_ch23_367-400.indd 390 05/09/12 10:49 AM

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only 3 The matrix is transposed. This means that the elements in the columns of the matrix become the

only 3 The matrix is transposed. This means that the elements in the columns of the matrix become the


C 3 U 5 K

C 5 £1 22 33 1 224 24 3

≥

Step 1 Solve for C 2 1 . Find the matrix of minors.

¶

2 124

22 32 23

4

22 32 23

4

1242222

24

332 21

4

332 21

4

22242222

1

3222 21

3

3222 21

3

22 12 ¶£(1 3 3 ) 2 (22 3 24 ) (3 3 3 ) 2 (22 3 4 ) ( 3 3 24 ) 2 (1 3 4)

(22 3 3 ) 2 ( 3 3 24 ) (1 3 3 ) 2 ( 3 3 4 ) (1 3 24 ) 2 (22 3 4 )(22 3 22 ) 2 ( 3 3 1) (1 3 22 ) 2 ( 3 3 3 ) (1 3 1) 2 (22 3 3 )

≥£25 17 216

6 29 41 211 7

≥

Step 2 Modify according to the matrix of coefficients. £1 2 1

2 1 2

1 2 1

≥ £25 217 21626 29 24

1 11 7≥

Step 3 Transpose the matrix. £ 25 26 1217 29 11216 24 7

≥

Step 4 Divide this matrix by the determinant | C| of the original matrix to obtain C 2 1 .

0C 0 5 1 2 1 22

24 322(22 )

23 224 3

2 13

23 14 24

2 5 1 3 (1 3 3 ) 2 (22 3 24) 2 (22 ) 3 ( 3 3 3 ) 2 (22 3 4 ) 1 3 3 ( 3 3 24 ) 2 ( 1 3 4 )

5 219

C21 5

1219

£ 25 26 1217 29 11216 24 7

≥ Step 5 Multiply both sides of the matrix equation by the inverse matrix.

C21 3 C 3 U 5 C21 3 K

U 5 C21 3 K

£xyz≥ 5

C21

£ 42723≥

£xyz≥ 5

1219

£ 25 26 1217 29 11216 24 7

≥ £ 42723≥

continued

alL70772_ch23_367-400.indd 391 05/09/12 10:49 AM

Sample

Divide this matrix by the determinant |

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Divide this matrix by the determinant | C

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C| of the original matrix to obtain

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| of the original matrix to obtain

2

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Exercises 23.11 1 a Express the simultaneous equations

2x 2 3y 5 9

5x 2 7y 5 22

as a single matrix equation.

b Solve the above equations using matrices, given that the inverse of the matrix ¢2 23

5 27≤ is ¢27 3

25 2≤ .

2 You are given that if A 5 ¢a b

c d≤ then A21 5

10A 0 5 ¢ d 2b

2c a≤ .

a If P 5 ¢3 2

5 4≤ :

i evaluate | P|

ii write down the matrix P 2 1

b Use the result for P 2 1 above to solve the system of equations

b3x 1 2y 5 4

5x 1 4y 5 10

3 If M 5 ¢4 3

1 22≤ and N 5 ¢2 3

1 24≤ :

a find the product MN

b write down the matrix M 2 1

c use the result of a above to solve the simultaneous equations below, showing each step

of the working:

b4x 1 3y 5 7

x 2 2y 5 10

4 For the following systems of simultaneous equations:

i express the equations in matrix form ii find the determinant

iii find the inverse matrix iv solve the equations

5

1219

£ (25 3 4 ) 1 (26 3 27 ) 1 (1 3 23 )(217 3 4 ) 1 (29 3 27 ) 1 (11 3 23 )(216 3 4 ) 1 (24 3 27 ) 1 (7 3 23 )

≥ 5

1219

£ 19238257

≥ 5 £21

23≥

x 5 21, y 5 2, z 5 3

continued

alL70772_ch23_367-400.indd 392 05/09/12 10:49 AM

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Solve the above equations using matrices, given that the inverse of the matrix

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Solve the above equations using matrices, given that the inverse of the matrix Solve the above equations using matrices, given that the inverse of the matrix

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Solve the above equations using matrices, given that the inverse of the matrix

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Solve the above equations using matrices, given that the inverse of the matrix only


a 2 3b 5 21

4a 2 11b 5 22

a 2 b 5 6

2a 1 3b 5 2

3x 2 y 5 6

22x 1 y 5 2

4p 1 4q 5 8

5p 2 3q 5 2

x 2 3y 5 4

4x 1 8y 5 6

5 The inverse of £4 21 1

3 22 21

1 21 21

≥ is £ 1 22 3

2 25 7

21 3 25

≥ . Use this result to solve the system

of equations c 4a 2 b 1 c 5 15

3a 2 2b 2 c 5 13

a 2 b 2 c 5 3

6 If P 5 £ 11 25 23

28 4 2

27 3 1

≥ and Q 51

2 £ 1 2 21

3 5 21

22 21 22

≥ :

a write down the matrix PQ

b write the system of equations below as a single matrix equation

c 11x 2 5y 2 3z 5 212

28x 1 4y 1 2z 5 10

27x 1 3y 1 z 5 10

c use the result of a above to solve the simultaneous equations in b using matrices.

7 If P 5 £ 3 24 2

22 1 0

21 1 0

≥ , Q 5 £1 2 22

2 5 24

3 7 25

≥ , R 5 £ 3 24 2

22 1 0

21 21 1

≥ :

a write down the matrices (i) PQ (ii) PR (iii) QR

b which two of these matrices are inverses of each other?

c express the simultaneous equations c a 1 2b 2 2c 5 3

2a 1 5b 2 4c 5 7

3a 1 7b 2 5c 5 8

as a single matrix equation

d solve the equations in c using matrices.

8 Solve the following systems of simultaneous equations using matrices:

a c 3x 1 2y 2 2z 5 23

2x 1 2y 2 z 5 1

24x 2 3y 1 2z 5 0

given that the inverse of £ 3 2 22

2 2 21

24 23 2

≥ is £21 22 22

0 2 1

22 21 22

≥

b c 2a 2 4b 2 c 5 0

5a 2 10b 2 3c 5 1

15a 2 29b 2 9c 5 5

given that if K 5 £ 2 24 21

5 210 23

15 229 29

≥, then K21 5 £3 27 2

0 23 1

5 22 0

≥

c c2p 1 3q 1 3r 5 22

3p 1 5q 1 5r 5 24

5p 1 3q 1 4r 5 0

given that £2 3 3

3 5 5

5 3 4

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13 27 21

216 9 1

≥ 5 £1 0 0

0 1 0

0 0 1

≥

a b c

d e

continued

alL70772_ch23_367-400.indd 393 05/09/12 10:50 AM

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write the system of equations below as a single matrix equation write the system of equations below as a single matrix equation only













write the system of equations below as a single matrix equation


d c2x 1 3y 1 2z 5 1

3x 1 4y 1 3z 5 1

5x 1 y 1 4z 5 21

given that £2 3 2

3 4 3

5 1 4

≥21

5 £ 13 210 1

3 22 0

217 13 21

≥

e c23x 1 5y 2 35z 5 22

13x 1 3y 2 20z 5 21

19x 1 4y 2 29z 5 22

given that £7 25 25

3 2 25

5 23 24

≥ £23 5 235

13 3 220

19 4 229

≥ 5 £1 0 0

0 1 0

0 0 1

≥

f c a 1 2b 1 3c 5 22

2a 1 3b 1 4c 5 0

3a 1 b 2 2c 5 17

given that £1 2 3

2 3 4

3 1 22

≥ £210 7 21

16 211 2

27 5 21

≥ 5 I

g c 2a 1 2b 5 26

3a 1 2b 1 c 5 0

7a 1 5b 1 2c 5 21

given that if M 5 £21 26 3

1 4 22

1 11 25

≥ , then M21 5 £2 3 0

3 2 1

7 5 2

≥

9 For the following systems of simultaneous equations:

i express the equations in matrix form

ii find the determinant

iii find the inverse matrix

iv solve the equations

a c2x 2 2y 2 z 5 1

x 2 5y 2 3z 5 2

x 1 6y 1 4z 5 23

b c2a 1 3b 1 2c 5 24

2a 1 2b 1 2c 5 22

5a 1 b 1 4c 5 26

c c p 1 q 1 r 5 4

2p 2 2q 1 r 5 5

2p 1 2q 1 4r 5 6

d c 2x 1 2y 2 2z 5 22

22x 1 2y 2 z 5 1

24x 2 3y 1 3z 5 0

e c4a 1 2b 2 4c 5 24

4a 2 5b 1 2c 5 22

5a 2 2b 2 c 5 25

continued

23.12 3 3 3 DETERMINANTS: DEFINITION AND EVALUATION 3 a b c

d e f

g h i

3 is a square array of ‘elements’ having three rows and three columns. It is called a

‘3 3 3 determinant’ or a ‘determinant of third order’.

The value of a second-order determinant may be defined to provide a shorthand method of

solving two simultaneous equations in two unknowns. The value of a third-order determinant

is defined so that it provides a shorthand method of solving three simultaneous equations in

three unknowns.

The value of the above determinant is defined as aei 1 bfg 1 cdh 2 gec 2 hfa 2 idb. There are

several ways of obtaining this result without having the very difficult task of committing it to memory.

We will use the method called the Rule of Sarrus. This method is the simplest but it applies only to

alL70772_ch23_367-400.indd 394 05/09/12 10:50 AM

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determinants of order three. (Later, when you study determinants of higher orders, you will learn other

methods and ones that are easier to program for a computer.)

The Rule of Sarrus 1 Write down the determinant, repeating the first two columns.

2 Obtain the products on the diagonals as shown below.

3 Add the lower products and add the upper products.

4 Subtract the sum of the upper products from the sum of the lower products.

Follow the application of this rule as we apply it to the general determinant 3 a b c

d e f

g h i

3 1

a b c a b

d e f d e

g h i g h

2, 3

gec hfa idb

aei bfg cdh

a

d

g

b

e

h

c

f

i

a

d

g

b

e

h

(Sum 5 aei 1 bfg 1 cdh)

(Sum 5 gec 1 hfa 1 idb)

4 Value: ( aei 1 bfg 1 cdh ) 2 ( gec 1 hfa 1 idb )

Example

Evaluate: 3 2 0 2321 4 0

3 1 22

3 Method:

Value: 5 (213 ) 2 (236 )

5 23

–36 0 0

0

4

1

–3

0

–2

2

–1

3

0

4

1

(Sum 5 236 )

(Sum 5 213 )

Hint: When copying down a determinant be very careful to include any negative signs. Check your copy before working on it. If you copied it row by row, check it column by column. It is very annoying to work on data that is later discovered to have been copied down incorrectly.

alL70772_ch23_367-400.indd 395 05/09/12 10:50 AM

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Exercises 23.12 1 Evaluate: ( Note: All the elements are exact numbers.)

a 32 1 22

3 0 21

5 0 1

3 b 3 4 3 5

0 5 0

24 4 25

3 c 3 1 1 0

1 21 2

21 1 2

3 d 3 1 2 3

2 23 28

25 22 1

3 e 33 1 2

0 4 1

2 3 2

3 f 3 0 t n

2t 0 x

2n 2x 0

3 g 342 28 62

36 29 91

37 30 47

3 h 3 20 224 28

231 47 264

83 251 286

3 2 In each case below, evaluate the pronumeral:

a 3 x 2x 3x

1 2 0

0 1 3

3 5 9 b 31 x 2

2 0 x

3 1 1

3 5 10 c 32 n 1

1 n 2

1 n 1

3 5 5

d 3 t t 1

4 1 0

22 t t

3 5 3 e 3 1 0 k

21 2 2

k 2 0

3 5 321 k 6

1 0 22

22 1 k

3 23.13 SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS

USING 3 3 3 DETERMINANTS A system of three linear equations in three unknowns may be solved algebraically by the

elimination method.

However, this method is usually very tedious and the equations are more easily solved

using determinants.

The general equations are: ca1x 1 b1y 1 c1z 5 d1

a2x 1 b2y 1 c2z 5 d2

a3x 1 b3y 1 c3z 5 d3

If we solve these equations by elimination we obtain the solution:

x 5^x

^, y 5

^y

^, z 5

^z

^

where: 5 3 a1 b1 c1

a2 b2 c2

a3 b3 c3

3 , which is the determinant formed by using the coeffi cients on the

left-hand side of the equation

^x 5 3 d1 b1 c1

d2 b2 c2

d3 b3 c3

3 , which is the same determinant but with the coeffi cients of x replaced

by the constants (i.e. the numbers on the right-hand side)

alL70772_ch23_367-400.indd 396 05/09/12 10:50 AM

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23.13 SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS

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23.13 SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS

3 DETERMINANTS

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3 DETERMINANTS

A system of three linear equations in three unknowns may be solved algebraically by the

Sample

A system of three linear equations in three unknowns may be solved algebraically by the

elimination method.

Sample

elimination method.

However, this method is usually very tedious and the equations are more easily solved Sample

However, this method is usually very tedious and the equations are more easily solved

using determinants. Sample

using determinants.

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^y 5 3 a1 d1 c1

a2 d2 c2

a3 d3 c3

3 , which is n again but with the coeffi cients of y replaced by

the constants

^z 5 3 a1 b1 d1

a2 b2 d2

a3 b3 d3

3 , which is n again but with the coeffi cients of z replaced by

the constants.

Example

Given: c x 2 2y 1 3z 5 4 3x 1 y 2 2z 5 27 4x 2 4y 1 3z 5 23

^ 5 31 22 33 1 224 24 3

3 5 (217 ) 2 25 219

12 8 –18

3 16 –36

1

3

4

–2

1

–4

3

–2

3

1

3

4

–2

1

–4

(2)

(–17)

Solution

x 5^x

^5

19219

5 21

y 5^y

^5

238219

5 2

z 5^z

^5

257219

5 3

^x 5 3 4 22 327 1 2223 24 3

3 5 ( 84 ) 2 ( 65 )5 19

–9 32 42

12 –12 84

4

–7

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–2

1

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3

–2

3

4

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–2

1

–4

(65)

(84)

^y 5 31 4 33 27 224 23 3

3 5 (280 ) 2 (242 )5 238

–84 6 36

–21 –32 –27

1

3

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4

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1

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4

4

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–3

(–42)

(–80)

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3 5 ( 5 ) 2 ( 62 )5 257

16 28 18

–3 56 –48

1

3

4

–2

1

–4

1

3

4

(62)

(5)

4

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1

–4

alL70772_ch23_367-400.indd 397 05/09/12 10:50 AM

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Points to note

■ After much practice, the value of a determinant can be found on a calculator, using the Rule of Sarrus, showing no intermediate results. The diagonal products can be summed using the M 1 and M 2 keys. However, for the time being, you are advised to write down the product of each diagonal before adding it to the calculator memory, as in the examples given. This allows you to check your work more easily and also allows for credit to be given in an examination for knowledge of the method even if an error is made during the computation.

■ When you have solved a set of simultaneous equations, check your result by substituting your values back into the original equations. You will be surprised how often careless errors are made during a long series of calculations.

■ When using a calculator, always work with all the signifi cant fi gures in the data. State your result to the appropriate number of signifi cant fi gures, that is, the number required or justifi ed.

Hints: a Before using the Rule of Sarrus to solve a set of simultaneous equations, make sure that:

i all the equations have the pronumerals in the same order ii all the pronumerals are on the left-hand sides of the equations and all the constants are on

the right-hand sides iii if a pronumeral is absent from an equation, write it in with a zero coefficient—for example,

if there is no y in an equation, write it in as 0 y. b When copying down an array to work on, be careful to place the elements in a neat rectangle

so that the diagonal elements are approximately in straight lines. An untidy array leads to errors in computation.

c Before working on your arrays, check that you have made no errors, especially by omitting any negative signs. Discover any errors before you start to multiply.

d When multiplying out diagonals, either mentally or with a calculator, ignore any negative signs present. Decide after the multiplication whether the product should be positive or negative.

Exercises 23.13 1 Solve for the pronumerals using determinants (do not use a calculator):

a c 2x 1 y 1 3z 5 4

3x 2 y 2 4z 5 5

4x 1 3y 1 2z 5 1

b c2p 1 4q 1 3r 1 8 5 0

p 1 3q 2 2r 1 2 5 0

3p 2 5q 2 4r 2 4 5 0

c c3x 1 2y 1 4z 5 27

2y 2 3z 1 4x 5 6

4z 2 5x 2 2y 5 27

d c4n 2 3p 1 5t 5 23

3n 1 4p 5 26

5p 2 4t 5 24

alL70772_ch23_367-400.indd 398 05/09/12 10:50 AM

Sample

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from an equation, write it in with a zero coefficient—for example,

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from an equation, write it in with a zero coefficient—for example, in an equation, write it in as 0

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in an equation, write it in as 0 y

Sample

y in an equation, write it in as 0 y in an equation, write it in as 0

Sample

in an equation, write it in as 0 y in an equation, write it in as 0 When copying down an array to work on, be careful to place the elements in a neat rectangle

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When copying down an array to work on, be careful to place the elements in a neat rectangle so that the diagonal elements are approximately in straight lines. An untidy array leads to

Sample

so that the diagonal elements are approximately in straight lines. An untidy array leads to

Before working on your arrays, check that you have made no errors, especially by omitting any

Sample

Before working on your arrays, check that you have made no errors, especially by omitting any negative signs. Discover any errors

Sample

negative signs. Discover any errors before

Sample

before

When multiplying out diagonals, either mentally or with a calculator, ignore any negative signs

Sample

When multiplying out diagonals, either mentally or with a calculator, ignore any negative signs present. Decide

Sample

present. Decide after

Sample

after the multiplication whether the product should be positive or negative.

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the multiplication whether the product should be positive or negative. after the multiplication whether the product should be positive or negative. after

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after the multiplication whether the product should be positive or negative. after

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Exercises 23.13 Sample







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ple


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ple

Solve for the pronumerals using determinants (do not use a calculator):

only

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only to solve a set of simultaneous equations, make sure that:

all the equations have the pronumerals in the

onlyall the equations have the pronumerals in the same order

onlysame order

all the pronumerals are on the left-hand sides of the equations and all the constants are on only

all the pronumerals are on the left-hand sides of the equations and all the constants are on

from an equation, write it in with a zero coefficient—for example, only

from an equation, write it in with a zero coefficient—for example,


c c2(3.7k 2 1.4t ) 1 3(4.3t 1 2.6w) 1 8.7 5 0

4(2.3k 1 1.7w) 2 5.3w 2 6.1 5 0

3(3.9k 1 4.2t ) 5 0

3

1 m 1 m 1 m

2 kN

F

P

2 F3

F1

37° 53°

The equations for equilibrium of this beam are:

Vertical forces: 0.6 F 2 2 2 1 0.8 F 3 5 0

Horizontal forces: F 1 1 0.8 F 2 2 0.6 F 3 5 0

Moments about point P: 0.6 F 2 2 4 1 2.4 F 3 5 0

Using determinants, solve for F 1 , F 2 and F 3 , stating the results correct to 3 significant figures.

Explain the meaning of any negative values.

4

Applying Kirchhoff’s laws to this network:

4I1 1 2( I1 2 I2) 5 12

2( I2 2 I1) 1 3( I2 2 I3) 5 6

3( I3 2 I2) 1 4I3 5 12

Use determinants to solve for I 1 , I 2 and I 3 , stating their values correct to 3 significant figures.

2 Solve for the pronumerals using a calculator:

a c37x 1 49y 1 76z 5 98

68y 2 34x 2 93z 5 136

82z 2 36y 1 29x 5 272

b c0.002x 1 0.003y 1 0.001z 5 0.01

0.3x 2 0.4y 1 0.2z 5 0.8

4000x 1 5000y 2 3000z 5 4000

Hint: In b, multiply or divide both sides of each equation by a constant so as to obtain simpler numbers.

State results correct to 3 significant figures.

12 VI1 I2 I3

6 V

12 V

2 3

4

4

continued

alL70772_ch23_367-400.indd 399 05/09/12 10:50 AM

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5 A developer receives council permission to divide his land of area 44 ha into 100 blocks, the only

areas allowed for a block being 2 ha, 0.5 ha and 0.2 ha. He prices the 2 ha blocks at $100 000

each, the 0.5 ha blocks at $40 000 each and the 0.2 ha blocks at $20 000 each. He sells all the

blocks, the total gross income from the sales being $3 200 000. How many blocks of each size

did he sell?

6 A firm has a stock of three different bronze alloys. Alloy A consists of 95% copper, 3% tin and

2% zinc. Alloy B consists of 90% copper, 9% tin and 1% zinc. Alloy C consists of 80% copper,

15% tin and 5% zinc. How many kilograms of each of these alloys must be melted and mixed in

order to produce 100 kg of a new alloy that consists of 87% copper, 9.6% tin and 3.4% zinc?

continued

alL70772_ch23_367-400.indd 400 05/09/12 10:50 AM

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order to produce 100 kg of a new alloy that consists of 87% copper, 9.6% tin and 3.4% zinc?order to produce 100 kg of a new alloy that consists of 87% copper, 9.6% tin and 3.4% zinc?

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