algebra probs

8/16/2019 Algebra Probs

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MATH1151Mathematics for Actuarial Studies

and Finance 1A

ALGEBRA PROBLEMS

Semester 1 2016

Copyright 2016 School of Mathematics and Statistics, UNSW


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Contents

Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Syllabus and lecture timetable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Problem schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Test schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 INTRODUCTION TO VECTORS 1

1.1 Vector quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Vector quantities and Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Rn and analytic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.4 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.5 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.6 Vectors, Matrices/Arrays and MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . 1

Problems for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 LINEAR EQUATIONS AND MATRICES 11

2.1 Introduction to linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Systems of linear equations and matrix notation . . . . . . . . . . . . . . . . . . . . 11

2.3 Elementary row operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Solving systems of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Deducing solubility from row-echelon form . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 Solving Ax = b for indeterminate b . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.7 General properties of the solution of Ax = b . . . . . . . . . . . . . . . . . . . . . . 11

2.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.9 Matrix reduction and MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11


3 MATRICES 23

3.1 Matrix arithmetic and algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 The transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 The inverse of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Some applications of matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . . 23

3.6 Matrices and MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23


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4 VECTOR GEOMETRY 33

4.1 Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 The dot product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Applications: orthogonality and projection . . . . . . . . . . . . . . . . . . . . . . . . 334.4 The cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.5 Scalar triple product and volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.6 Planes in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.7 Projections and least squares approximations . . . . . . . . . . . . . . . . . . . . . . 334.8 Vector Geometry and MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


5 INTRODUCTION TO PROBABILITY AND STATISTICS 43

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Some preliminary set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.4 Discrete random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.5 Continuous random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.6 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.7 Sums and means of random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.8 Approximations to the Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . 435.9 Probability and MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43


ANSWERS TO SELECTED PROBLEMS 59Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76


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ALGEBRA SYLLABUS AND LECTURE TIMETABLE

The algebra course for MATH1151 is based on the MATH1151 Algebra Notes, which are essentialreading and must be brought to all algebra tutorials. There is very little overlap between thissyllabus and the algebra specified in the NSW HSC curriculum. The computer package Matlabwill be used in the MATH1151 algebra course. An approximate lecture timetable is given below.The lecturer will try to keep to this timetable, but variations might be unavoidable, especially dueto public holidays.

Chapter 1. Introduction to Vectors

Lectures 1–4

Vector quantities. Rules for addition and scalar multiplication of geometric vectors.

Brief mention of matrices for Matlab applications. Addition of vectors and multiplication by scalars.(Section 1.1)Vector quantities and Rn. (Section 1.2)Analytic geometry and other applications. (Section 1.3)Points, line segments and lines. Displacements. Lines in R2, R3, and Rn. (Section 1.4)Parametric vector equations for planes in Rn. The linear equation form of a plane. (Section 1.5)

Chapter 2. Linear Equations and Matrices

Lectures 5–8

Introduction to systems of linear equations. Solution of 2 × 2 and 2 × 3 systems and geometricalinterpretations. (Section 2.1)Matrix notation. (Section 2.2)Elementary row operations, elementary matrices. (Section 2.3)Solving systems of equations via Gaussian elimination.(Section 2.4)Deducing solubility from row-echelon form. (Section 2.5)Solving systems with indeterminate right hand side. (Section 2.6)General properties of solutions to Ax = b. (Section 2.7)Applications in Actuarial Studies, Finance and Commerce. (Section 2.8)

Chapter 3. Matrices

Lectures 9–11

Operations on matrices. (Section 3.1)Transposes. (Section 3.2)Inverses. (Section 3.3)Determinants. (Section 3.4)Applications of matrix multiplication. (Section 3.5)


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Chapter 4. Vector Geometry

Lectures 12–15

Length, distance between points. (Section 4.1)Angles and dot product in R2, R3, Rn. (Section 4.2)Orthogonality and orthonormal basis, pro jection of one vector on another. Relationship betweencoordinates of a vector and projections of the vector on orthonormal basis vectors. (Section 4.3)The cross product: definition and arithemetic properties, geometric interpretation of cross productas perpendicular vector and area (Section 4.4)Scalar triple products, determinants and volumes (Sections 4.5)Equations of planes in R3: the parametric vector form, linear equation (Cartesian) form and point-normal form of equations, the geometric interpretations of the forms and conversions from one formto another. (Section 4.6)Projections and least-squares approximations (Section 4.7)

Chapter 5. Probability

Lectures 16–24

Introduction to probability and statistics. (Section 5.1)Preliminary set theory. (Section 5.2)Axiomatic probability, sample spaces, conditional probability, Bayes rule, independent events. (Sec-tion 5.3)Discrete random variables (uniform, binomial, Poisson, geometric). Mean and variance of a discreterandom variable. (Section 5.4)Continuous random variables (uniform, negative exponential). Cumulative distribution functions.

Mean and variance of a continuous random variable. (Section 5.5)The normal distibution. The standard normal distribution. Evaluating normal probability inte-grals. Conversion from general normal distributions to standard normal distributions. Applicationsof the normal distribution. Estimation of probabilities. (Section 5.6)The sampling distribution for the mean and the central limit theorem. Sums of random variables.(Section 5.7)Approximations to the binomial distribution by the normal distribution and by the Poisson distri-bution. (Section 5.8)

ALGEBRA PROBLEM SETS

The Algebra problems are located at the end of each chapter of the Algebra Notes booklet. They

are also available from the course module on the UNSW Moodle server. Some of the problems arevery easy, some are less easy but still routine and some are quite hard. To help you decide whichproblems to try first, each problem is marked with an [R] or an [H]. The problems marked [R] forma basic set of problems which you should try first. Problems marked [H] are harder and can be leftuntil you have done the problems marked [R]. You do need to make an attempt at the [H] problemsbecause problems of this type will occur on tests and in the exam. If you have difficulty with the[H] problems, ask for help in your tutorial.

The problems marked [X] are intended for students in MATH1141 – they relate to topics whichare only covered in MATH1141 and are included only for interest. There are a number of questionsmarked [M], indicating that Matlab is required in the solution of the problem.


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Chapter 1

INTRODUCTION TO VECTORS

1.1 Vector quantities

1.2 Vector quantities and Rn

1.3 Rn and analytic geometry

1.4 Lines

1.5 Planes

1.6 Vectors, Matrices/Arrays and MATLAB

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2 PROBLEMS FOR CHAPTER 1

Problems for Chapter 1

Problems 1.1

1. [R] Given that ABC, DEF, and OGH are equally spaced parallel lines, as are ADO, BEGand CF H. P is the mid point of AD .

A

D

O

B C

E

G

F

H

P

If −−→OH = h and −→OA = a, express the following in terms of a and h.

a) −−→

OC , b) −−→

HA, c) −−→

GC, d) −−→

OP , e) −−→

GP .

2. [R] Simplify

a) −−→

AB − −−→OB + −→OA, b) −−→AB − −−→CB + 3−−→DA + 3−−→CD.

3. [R] Express each of the following in terms of a and b.

a) 3(2a + b)

−2(5a

−b),

b) 2( p a + q b) + 3(r a − s b) where p, q, r, s ∈ R.

4. [R] Let AB C be a triangle with −→OA = a,

−−→OB = b,

−−→OC = c where O is the origin.

a) If M is the midpoint of the line segment AB and P is the midpoint of the line segment

CB express the vectors −−→OM and

−−→OP in terms of a, b, and c.

b) Show that −−→M P is parallel to

−→AC and has half its length.

5. [H] Given a convex quadrilateral ABCD, prove, using vectors, that the quadrilateral formedby joining the midpoints of AB, B C , C D, and DA is a parallelogram.

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c) u =

11

12

, v =

−12

10

, w =

21

31

;

d) u =

−13

5

, v =

102

−3

, w = 0

0

;

e) u = 2i + 3 j − 2k, v = i − 2 j + k, w = −i + j − k.

10. [R] A car travels 3km due North then 5km Northeast. Use coordinate vectors to find thedistance and direction from the starting point.

11. [R] Solve Problems 6(a) and (b) using coordinate vectors.

12. [R] Suppose that v =

a1b1

c1

and w =

a2b2

c2

are vectors in R3; λ and µ are real numbers.

Prove the scalar distributive law (λ + µ)v = λv + µv and the vector distributive lawλ(v + w) = λv + λw.

13. [H] Prove the associative law of vector addition in Rn. (Proposition ?? on page ??).

14. [H] Prove Proposition ?? on page ??.

Problems 1.3

15. [R] Let v =

23

and w =

−11

. Draw coordinate axes and mark in the points whose

coordinate vectors are v, −v, w, v + w, 2v and v − w.

16. [R] Given the following points A, B, C and D, are the vectors −−→AB and

−−→CD parallel?

a) A = (1, 2, 3), B = (−2, 3, 4), C = (−3, −4, 7), D = (4, −6, −9);b) A = (3, 2, 5), B = (5, −3, −6), C = (−2, 3, 7), D = (0, −2, −4);c) A = (12, −4, 6), B = (2, 6, −4), C = (5, −2, 9), D = (0, 3, 4).

Do any of these sets of 4 points form a parallelogram?

17. [R] Prove that A(1, 2, 1), B(4, 7, 8), C (6, 4, 12) and D(3, −1, 5) are the vertices of a parallel-ogram. Draw and label the parallelogram.

18. [R] Show that the points A(1, 2, 3), B(3, 8, 1), C (7, 20, −3) are collinear.

19. [R] Show that the points A(

−1, 2, 1), B(4, 6, 3), C (

−1, 2,

−1) are not collinear.

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CHAPTER 1. INTRODUCTION TO VECTORS 5

20. [R] Show that the points A, B,C in R3 with coordinate vectors

a =10

3

, b = 014

and c = 6−5−2

are collinear.

21. [H] If A(−1, 3, 4), B(4, 6, 3), C (−1, 2, 1) and D are the vertices of a parallelogram, find all thepossible coordinates for the point D .

22. [H] Consider three non-collinear points D, E, F in R3 with coordinate vectors d, e and f .There are exactly 3 points in R3 which, taken one at a time with D, E and F, form aparallelogram. Calculate vector expressions for the three points.

23. [R] Let A = (2, 3, −1) and B = (4, −5, 7). Find the midpoint of A and B. Find the point Qon the line through A and B such that B lies between A and Q and B Q is three times aslong as AB .

24. [R] The coordinate vectors, relative to the origin O, of the points A and B are respectively aand b. State, in terms of a and b, the position vector of the point T which lies on AB

and is such that −→AT = 2

−→T B.

25. [R] List the standard basis vectors for R5.

26. [R] For each of the following vectors, find its length and find a vector of length one (“unit”vectors) parallel to it.

a =

4−4

2

, b =

2103

, c =

401

−20

.

27. [R] Find the distances between each of the following pairs of points with coordinate vectors:

a)

8−4

2

,

−61

0

; b)

11

1

,

5−7

−7

; c)

3014

,

−2613

.

28. [R] A triangle has vertices A, B and C which have coordinate vectors

41

7

,

7−4

6

and

62

8

respectively. Find the lengths of the sides of the triangle and deduce that the triangle isright-angled.

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29. [H] Construct a cube in R3 with the length of each edge 1. Show that the face diagonal haslength

√ 2 and the long diagonal

√ 3. Try to generalise this idea to R4 and show that there

are now diagonals of length √ 2, √ 3 and 2. How many vertices does a 4-cube have?30. [H] Find 10 vectors in R10, each pair of which is 5

√ 2 apart. Can you now find an 11th such

vector?

Problems 1.4

31. [R] Find the coordinate vector for the displacement vector −−→AB and parametric vector forms

for the lines through the points A and B with coordinates

a) A (1, 2), B (2, 7); b) A (1, 2,

−1), B (

−1,

−1, 5);

c) A (1, 2, 1), B (7, 2, 3); d) A (1, 2, −1, 3), B (−1, 3, 1, 1).

32. [R] Does the point (3, 5, 7) lie on the line x =

−13

6

+ λ

42

1

?

33. [R] Find parametric vector forms for the following lines in R2:

a) y = 3x + 4; b) 3x + 2y = 6; c) y = −7x;d) y = 4; e) x = −2.

In each case indicate the direction of the line and a point through which the line passes.

34. [R] Find a parametric vector form and a Cartesian form for each of the following lines

a) through the points (−4, 1, 3) and (2, 2, 3);

b) through (1, 2, −3) parallel to the vector 4−5

6

;

c) through (1, −1, 1) parallel to the line joining the points (2, 2, 1) and (7, 1, 3);d) through (1, 0, 0) parallel to the line joining the points (3, 2, −1) and (3, 5, 2).

35. [R] Let A, B,P be points in R3 with position vectors

a =

7−2

3

, b =

1−5

0

and p =

1−1

2

.

Let Q be the point on AB such that AQ = 2

3 AB.

a) Find q, the position vector of Q.

b) Find the parametric vector equation of the line that passes through P and Q.

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36. [R] Decide whether each of the following statements is true or false.

a) The lines y = 3x − 4 and x = 21 + λ 412 are parallel.b) The lines x =

3−1

+ λ

64

and 2x + 3y = 8 are parallel.

c) The lines x =

4−1

2

+ λ

102

8

and x + 10

5 = y − 7 = z + 3

4 are parallel.

d) The line x =

3−2

7

+ λ

100

−4

and the line

x + 105

= z + 3−2 and y = −5

are parallel.

37. [H] Suppose A and B are points with coordinate vectors a and b, respectively. Write down aparametric vector form for

a) the line segment AB .

b) the ray from B through A.

c) all points P which lie on the line through A and B such that A is between P and B .

d) all points Q which lie on the line through A and B and are closer to B than A.

38. [H] Give a geometric interpretation of the following sets. In each set, λ ∈ R.

a) S =

x : x =

13

6

+ λ

−31

7

for 0 λ 1

.

b) S =

x : x =

1240

−7

+ λ

−25936

for − 1 λ 5

.

c) S =

x : x = λ

6−2

72

−15

+ (1 − λ)

0483

−54

for 0 λ 1

.

d) S =

x : x =

14

−62

+ λ

30

−15

for λ 0

.

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e) S =

x : x =

31

−4

+ λ

6−2

7

for |λ| 2

.

Problems 1.5

39. [R] Find a parametric vector form for the planes passing through the points

a) (0, 0, 0), (3, −1, 2), (1, 4, −6); b) (1, 4, −2), (2, 6, 4), (1, −10, 3).

40. [R] For each of the following sets of vectors, decide if the set is a line or a plane, give a point onthe line or plane, and give vectors parallel to the line or plane, i.e., geometrically describe

the sets.

a) S =

x : x = λ1

12

3

+ λ2

−23

4

for λ1, λ2 ∈ R

.

b) S =

x : x =

3124

+ λ1

−2132

+ λ2

4−2−6−4

for λ1, λ2 ∈ R

.

c) span

3

212

, −9

−6−3−6

.

d) S =

x : x =

12

3

+ y for y ∈ span

4−1

2

,

82

4

.

41. [R] Find parametric vector forms for the planes

a) through the point (1, 2, 3) parallel to 21

3 and

−12

−3;b) through the points (3, 1, 4), (−1, 2, 4), (6, 7, −2);c) through the points (−2, 4, 1, 6), (3, 2, 6, −1), (1, 4, 0, 0);

d) 4x1 − 3x2 + 6x3 = 12, where x =x1x2

x3

∈ R3;

e) 5x2 − 6x3 = 5, where x =

x1x2x3

∈ R3;

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f) through the point (1, 2, 3, 4) parallel to the lines x =

−31

24

+ λ

40

−45

and

x1 − 57

= x2 + 6

2 =

x3 − 2−3 =

x4 + 1

−5 .

42. [R] Find parametric vector forms to describe the following planes in R3.

a) x1 + x2 + x3 = 0. b) 3x1 − x2 + 4x3 = 12.c) x2 + 6x3 = −1. d) x3 = 2.

43. [H] Show that the line x = t213

a) lies on the plane 4x − 5y − z = 0, and b) is parallel to the plane 3x − 3y − z = 2.

44. [H] a) Find the intersection of the line x =

2 + t3 − t

4t

and the plane 2x + 3y + z = 16.

b) Find the intersection of the line x =

−12

3

+λ

2−3

4

and the plane 9x+4y−z = 0.

45. [H] a) Write the plane x =

−32

6

+ λ

24

0

+ µ

−10

3

in Cartesian form.

b) Write the plane x =

6−1

4

+ λ

−16

6

+ µ

21

0

in Cartesian form.

46. [H] Consider the line x − 3

−2 = y + 2

3 = z − 1 and the plane 2x + y + 3z = 23 in R3.

a) Find a parametric vector form for the line.

b) Hence find where the line meets the plane.

47. [H] Let ℓ be the line x − 6

5 =

y − 42

= z − 1

−2 in R3.

a) Express the line ℓ in parametric vector form.

b) Find the coordinates of the point where ℓ meets the plane 2x + y − z = 1.

48. [H] The following sets of points represent simple geometric figures in a plane. λ1 and λ2 arereal numbers. For each problem draw a sketch in the (λ1, λ2) plane and a second sketchin R2,R3 or R4 (!!) as appropriate. For each problem identify the geometric shape.

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a) S =

x : x =

01

+ λ1

12

+ λ2

23

for 0 λ1 1, 0 λ2 1

.

b) S =

x : x =

01

+ λ1

12

+ λ2

23

for 0 λ1 1, 0 λ2 λ1

.

c) S =

x : x = λ1

21

−2

+ λ2

4−2

3

for 0 λ1 6, 0 λ2 8

.

d) S =

x : x = λ1

21

−2

+ λ2

4−2

3

for 0 λ1 6, 0 λ2 λ1

.

e) S =x : x = λ1

21

−22

+ λ2

4

−2

3−1

for 0

λ1

6, 0

λ1

λ2 .

49. [H] Write down the sets of points corresponding to the following:

a) A “parallelogram” with the three vertices A(1, 3, 4, 2), B(−2, 1, 0, 5) and C (−4, 0, 6, 8).Hint: Look at Question 48 a), and assume B and C are adjacent to A.

b) The triangle with the three vertices given in part a) of this question.Hint: Look at Question 48 b).

c) All three parallelograms which have the three vertices given in part a).

50. [H] Given two planes in Rn, n 3:

x = a + s1u1 + s2u2 and x = b + t1v1 + t2v2,

for s1, s2, t1, t2 ∈ R. These two planes are said to be parallel if span(u1, u2) = span(v1, v2).Consider the pair of planes in R4 with equations

x = s1e1 + s2e2 and x = e4 + t1e2 + t2e3

for s1, s2, t1, t2 ∈ R, where {e1, e2, e2, e4} is the standard basis of R4.

Show that these form a pair of skew planes; that is, they are non-parallel and non-intersecting.

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Chapter 2

LINEAR EQUATIONS ANDMATRICES

2.1 Introduction to linear equations

2.2 Systems of linear equations and matrix notation

2.3 Elementary row operations

2.4 Solving systems of equations

2.5 Deducing solubility from row-echelon form

2.6 Solving Ax = b for indeterminate b

2.7 General properties of the solution of Ax = b

2.8 Applications

2.9 Matrix reduction and MATLAB

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Problems 2.1

1. [R] Find the solution set of each of the following linear equation.

a) 2x1 − 5 = 0 as an equation of one variable, then as an equation in two variables, andthen three variables.

b) x1 + 2x2 = 4 as an equation of two variables, then three variables.

c) 2x1 − 3x2 + x3 = 2 as an equation of three variables.

2. [R] Determine algebraically whether the following systems of equations have a unique solution,

no solution, or an infinite number of solutions. Draw graphs to illustrate your answers.

a) 3x1 + 2x2 = 69x1 + 6x2 = 36

b) 3x1 + 2x2 = 69x1 + 4x2 = 36

c) x1 − 5x2 = 56x1 − 30x2 = 30

3. [H] Find conditions on the coefficients a11, a12, a21, a22, b1, b2 so that the system of equations

a11x1 + a12x2 = b1a21x1 + a22x2 = b2

has a) a unique solution, b) no solution, and c) an infinite number of solutions.For simplicity, assume a11 = 0.

4. [H] Repeat the previous question with no simplifying assumptions. That is, find generalconditions which apply for all possible values of the coefficients.

5. [R] Find and geometrically describe the solutions for the following systems of linear equations.

a) x1 + 2x2 + 3x3 = 52x1 + 5x2 + 8x3 = 12

b) 4x1 + 5x2 − 2x3 = 168x1 + 10x2 − 4x3 = 20

c) 4x1 + 5x2 − 2x3 = 168x

1 + 10x

2 − 4x

3 = 32

6. [H] Prove algebraically that two distinct planes in R3 either intersect in a line or are parallelwith no points in common. Use a linear equation in three unknowns to represent a planein R3.

7. [R] Show that x1 = 2 − 2λ, x2 = λ, x3 = 3 + 2λ, where λ is any real number, satisfy thesystem of equations

x1 + 4x2 − x3 = −12x1 + 4x2 = 4

6x2 − 3x3 = −9

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CHAPTER 2. LINEAR EQUATIONS AND MATRICES 13

Problems 2.2

8. [R] Write each of the following systems of equations in vector form, as a matrix equationAx = b, and in the augmented matrix (A|b) form.a) 3x1 − 3x2 + 4x3 = 6

5x1 + 2x2 − 3x3 = 7−x1 − x2 + 6x3 = 8

b) x1 + 3x2 + 7x3 + 8x4 = −23x1 + 2x2 − 5x3 − x4 = 7

3x2 + 6x3 − 6x4 = 5

9. [R] Write the system of equations, the matrix equation and the augmented matrix form cor-

responding to the vector equation

x1

10

−67

+ x2

−36

−19

+ x3

06

−411

=

10−2

05

.

Problems 2.3

10. [R] For each of the following matrices, find the appropriate elementary row operations todescribe the transformation from one matrix to the next. Also continue the row reduction

until the matrix is in row echelon form.

a)

1 4 2 32 6 3 0

4 −2 4 4

→

1 4 2 30 −2 −1 −6

0 −18 −4 −8

,

b)

3 4 1 32 8 0 2

0 8 3 0

→

1 −4 1 11 4 0 1

0 8 3 0

.

Problems 2.4

11. [R] For each of the following augmented matrices do the following. Determine whether thematrix is in row-echelon form as defined in Section ??. If the matrix is in row-echelon form,identify the leading elements, leading rows, leading columns, and non-leading columns.

a)

3 2 1 100 4 2 8

0 0 −7 14

, b) 3 2 1 10 , c)

3 2 1 104 0 2 8

0 0 −7 14

,

d)

3 2 1 100 4 2 8

, e)

0 3 1 60 0 1 5

, f)

3 2 1 100 4 2 80 0 −7 140 0 0 0

,

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g)

3 2 1 100 4 2 8

0 0 −7 140 0 0 6

, h)

3 2 1 100 4 2 8

0 0 0 00 0 0 6

.

12. [R] Find the solutions to the following systems of equations. If possible give a geometricinterpretation of the solution.

a) 3x1 + 2x2 + x3 = 104x2 + 2x3 = 8

− 7x3 = 14b) 3x1 + 2x2 + x3 + x4 = 10

4x2 + 2x3 − 4x4 = 8

− 7x3 + 14x4 = 14

13. [R] For each of the following systems of equations, do the following:

i) Write down the corresponding augmented matrix.

ii) Use Gaussian elimination to transform the augmented matrix into row-echelon form.

iii) Solve each system of equations writing your answer in vector form.

a) x1 − 2x2 = 53x1 + x2 = 8

b) x1 − 2x2 − 3x3 = 32x1 + 4x2 + 10x3 = 14

c) x1 − 2x2 + 3x3 = 112x1 − x2 + 3x3 = 104x1 + x2 − x3 = 4

d) 2x1 − 2x2 + 4x3 = −33x1 − 3x2 + 6x3 = −45x2 + 2x3 = 9

e) x1 + 2x2 + 4x3 = 10−3x1 + 3x2 + 15x3 = 15−2x1 − x2 + x3 = −5

f) x1 − 4x2 − 5x3 = −62x1 − x2 − x3 = 23x1 + 9x2 + 12x3 = 30

g) x1 + 2x2 − x3 + x4 = 4x2 − x3 + x4 = 1

3x1 + 2x2 − 2x4 = 35x1 + 3x2 − x4 = 9

h) x1 + 2x2 − x3 + x4 = 4x2 − x3 + x4 = 1

3x1 + 2x2 − 2x4 = 3

14. [R] For each of the following augmented matrices, find a reduced row-echelon form. Then writedown all solutions of the corresponding system of equations and try to give a geometricinterpretation of the solutions.

a)

2 4 1 40 1 2 −20 0

−1 2

, b)

1 2 3 4 10 −1 5 6 20 0 1 7 3

.

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Problems 2.5

15. [R] For each of the following augmented matrices, without solving, decide whether or not thecorresponding system of linear equations has a unique solution, no solution or infinitelymany solutions.

a)

3 2 1 100 4 2 8

0 0 −7 14

, b)

3 2 1 100 4 2 80 0 −7 140 0 0 6

, c) 3 2 1 10 ,

d)

3 2 1 100 4 2 8

, e)

3 2 1 100 4 2 80 0 −7 00 0 0 0

.

16. [H] Determine which values of k, if any, will give a) a unique solution b) no solutionc) infinitely many solutions to the system of equations

x + y + kz = 23x + 4y + 2z = k2x + 3y − z = 1.

17. [H] For which values of λ do the equations

x + 2y + λz = 1

−x + λy − z = 0λx − 4y + λz = −1

have a) no solutions, b) infinitely many solutions, c) a unique solution?

18. [H] Consider the equation

1 2 3 00 2 2 −10 0 3 10 0 0 a

x1x2x3x4

=

50a

a + 2b

.

For what values of a and b does the equation havea) a unique solution, b) no solution,c) infinitely many solutions? d) In the case of (c), determine all solutions.

19. [H] You are an auditor for a company whose four executives make regular business trips onfour routes and you suspect that at least one of the executives has been overstating herexpenses. You don’t know how much it costs to travel each route, but you know that it isthe same for all the executives. You know the number of trips each executive made on eachroute in a certain period and you know the total expenses claimed by each executive forthis period. If the numbers of trips are as shown in the table below, do you have sufficientinformation to be sure that someone is cheating? State your reasoning clearly.

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Route1 2 3 4

Executive A 0 1 1 2Executive B 1 2 0 1Executive C 3 4 0 1Executive D 2 1 3 3

20. [H] P,Q ,R and S are four cities connected by highways which are labelled as shown in thediagram.

Q

P S

R

c

a

d

b

e

A hire car operator in P makes a note of the number of kilometres travelled by fivecustomers who made trips starting and ending at P . He knows that the routes travelledby the five customers were as follows: abdc abdea cddc cdbec aedbec

Can he determine the length of each of the five highways? State your reasoning clearly.

Problems 2.6

21. [R] For each of the following systems of linear equations, find x1, x2 and x3 in terms of b1, b2and b3.

a) x1 − 2x2 + 3x3 = b1x2 − 3x3 = b2

−2x1 + 3x2

− 2x3 = b3

b) 2x1 − 4x3 = b13x1 + x2 − 2x3 = b2

−2x1

− x2

− x3 = b3

22. [R] Show that the system of equations x + y + 2z = a, x + z = b and 2x + y + 3z = c areconsistent if and only if c = a + b.

23. [R] For the following systems, find conditions on the right-hand-side vector b which ensurethat the system has a solution.

a) 2x1 − 4x3 = b13x1 + x2 − 2x3 = b2

−2x1 − x2 = b3

b) x1 + x2 + 3x3 − x4 = b12x1 − x2 + 2x4 = b2

x1 − 2x2 − 3x3 + 3x4 = b33x2 + 6x3

− 4x4 = b4

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Problems 2.7

24. [R] Show that x =

72

0

+ λ

−20

1

, λ ∈ R are the solutions of

x1 − 2x2 + 2x3 = 32x1 − 6x2 + 4x3 = 2

−2x1 + 4x2 − 4x3 = −6

and that x = λ

−2

01

, λ ∈ R are the solutions of the corresponding homogeneous system

x1 − 2x2 + 2x3 = 02x1 − 6x2 + 4x3 = 0

−2x1 + 4x2 − 4x3 = 0

Problems 2.8

25. [R] Does the point (−3, 3, 6) lie on the plane

x =

21

−1

+ λ1

−12

4

+ λ2

32

1

?

26. [R] Is the vector

13

2

in span

−13

4

,

21

3

?

27. [R] Is the vector 11

412

in span3

−1

46

,4

−2

43

?

28. [R] Can

31

−24

be expressed as a linear combination of

10

−37

and

2−1

56

?

29. [R] Do the lines x =

213

+ λ1

132

and x =

1215

7

+ λ2

31

−2

intersect?

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30. [R] Is the vector

57

−1

parallel to the plane x =

21

3

+ λ1

12

1

+ λ2

35

1

for λ1, λ2 ∈ R?

31. [H] Show that the line x − 1

2 =

y

3 =

z + 1

−1 is parallel to the plane

x = λ1

11

0

+ λ2

01

−1

, λ1, λ2 ∈ R.

32. [H] Find the intersection (if any) of the line x = 0

181 + µ

2

−31 for µ ∈ R and the plane

x =

10

4

+ λ1

14

1

+ λ2

31

−2

for λ1, λ2 ∈ R.

33. [R] Find the intersection (if any) of the planes 8x1 + 8x2 + x3 = 35 and

x =

6−2

3

+ λ1

−21

3

+ λ2

11

−1

for λ1, λ2 ∈ R.

34. [H] Are the planes

x =

1−4

23

+ λ1

21

−27

+ λ2

−3152

for λ1, λ2 ∈ R

and

x =

2

−41

3

+ µ1

3

−12

4

+ µ2

−1

42

6

for µ1, µ2 ∈ Rparallel?

35. [R] Show that the 3 planes with Cartesian equations

x + 3y + 2z = 5

2x + y − z = 27x + 11y + 4z = 13

do not intersect at one point.

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36. [H] Consider the following system of equations

x + y−

z = 1

2x − 4y + 2z = 23x − 3y + z = 3

a) Use Gaussian elimination and back–substitution to find the solution(s), if any, of theabove equations.

b) Use your result in part a) to decide whether the three planes represented by theequations are parallel, intersecting in a straight line, intersecting at a point or havesome other configuration.

37. [R] Find a polynomial p(x) of degree 2 satisfying p(1) = 5, p(2) = 7, p(3) = 13.

38. [R] The total of the ages of my brother, my sister and myself is 140 years. I am seven timesthe difference between their ages (my sister is older than my brother) and in seven yearsI will be half their combined ages now. How old are we?

39. [R] In a trip to Asia a traveller spent $90 a day for hotels in Bangkok, $60 a day in Singaporeand $60 a day in Kuala Lumpur. For food the traveller spent $60 a day in Bangkok, $90a day in Singapore, and $60 a day in Kuala Lumpur. In addition the traveller spent $30a day in other expenses in each city. The traveller’s diary shows that the total hotel billwas $1020, total food bill was $960, and total other expenses were $420. Find the numberof days the traveller spent in each city, or show that the diary must be wrong.

40. [R] A dietician is planning a meal consisting of three foods. A serving of the first food contains5 units of protein, 2 units of carbohydrates and 3 units of iron. A serving of the secondfood contains 10 units of protein, 3 units of carbohydrates and 6 units of iron. A servingof the third food contains 15 units of protein, 2 units of carbohydrates and 1 unit of iron.How many servings of each food should be used to create a meal containing 55 units of protein, 13 units of carbohydrates and 17 units of iron?

41. [R] Assume 3 countries A, B , and C trade with one another and no-one else, that a commoncurrency is used and that each country’s total income comes from trade with the othersor sales to itself and nothing else.

A spends 34

, 18

, 18

of its income on goods from A, B , C respectively.

B spends 1

5, 3

5, 1

5 of its income on goods from A, B , C respectively.

C spends 1

4, 1

4, 1

2 of its income on goods from A, B , C respectively.

Find the (relative) income of each country.

42. [R] A simple economy is based on 3 commodities, grain, fuel and transportation. Productionof 1 unit of grain requires 1/4 unit of fuel and 1/3 unit of transportation, productionof 1 unit of fuel requires 1/2 unit of grain and 1/4 unit of transportation, while 1 unit

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of transportation requires 1/4 unit of grain and 1/4 unit of fuel. Find the input-outputmatrix for this situation and the gross output required for a net production of 516 units

of grain, 258 units of fuel and 129 units of transportation. (Calculator required)

43. [R] In a certain nation the economy is divided into three sectors, primary (agriculture, mining,etc.), secondary (manufacturing) and teriary (service industries). It has been found that

to produce one unit of primary products, it is necessary to use 1

6 unit of primary products,

1

8 unit of secondary products and

1

6 unit of tertiary products.

Similarly to produce one unit of secondary products it is necessary to use 1

3 unit of primary

products, 1

4 unit of secondary products and

1

2 unit of tertiary products. To produce one

unit of tertiary products it is necessary to use 112 unit of primary products, 14 unit of

secondary products and 1

3 unit of tertiary products. Suppose that in a year the nation

exports 210 units of primary products, 105 units of secondary products and 7 units of tertiary products.

Set up a Leontieff input-output model to mathematically describe the economy of thenation based on these three sectors. How much must each sector produce in a year toexactly satisfy the internal and external (export) demands of the economy? (MATLABrequired)

44. [H] A farmer owns a 12-hectare farm on which he grows wheat, oats and barley. Each hectareof cereal crop planted has certain requirements for labour, fertiliser and irrigation wateras shown in the following table.

Crop Labour Fertiliser Irrigation Water(hours per week) (kilograms) kilolitres)

Wheat (per hectare) 6 150 72Oats (per hectare) 6 100 48Barley (per hectare) 2 70 36

Amount available 48 700 612

Answer the following questions

a) Set up a linear equation model for the system.

b) Find the solution (if any) of the model.

c) Replace the equations by inequalities assuming that not all of the available land,labour time, fertiliser and irrigation water have to be used. Then introduce 4 new“slack” variables which represent the amounts of unused land, labour, fertiliser andirrigation water respectively.

d) Can you find any reasonable solutions for the systems of linear equations in (c), i.e.solutions in which the variables are non-negative?

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45. [H] In this problem we shall calculate the area of a spherical triangle. Consider the surfaceof a sphere of unit radius of area 4π. A great circle on a sphere is the intersection of

that sphere with a plane through the centre. If two great circles meet at antipodal pointsP, P ′ let the angle θ between them be the angle 0 < θ < π between the tangents to thetwo circles at P . (π − θ is also the angle between the two great circles). Finally define aspherical triangle to be the region bounded by 3 great circles meeting A,B, C with anglesα,β,γ .

a) The areas bounded by 2 great circles are called lunes. Show their areas are 2θ, 2θ,2(π − θ), 2(π − θ).

b) Show the surface of the sphere is divided by a spherical triangle into 8 regions equalin area in pairs.

c) Use parts a) and b) to set up a simple system of 4 linear equations in the 4 areas.

d) Hence show area AB C = α + β + γ − π.

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22 CHAPTER 2. LINEAR EQUATIONS AND MATRICES


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Chapter 3

MATRICES

3.1 Matrix arithmetic and algebra

3.2 The transpose of a matrix

3.3 The inverse of a matrix

3.4 Determinants

3.5 Some applications of matrix multiplication

3.6 Matrices and MATLAB

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Problems 3.1

1. [R] Given the matrices

A =

2 −3 43 2 −2

1 −1 3

, B =

−2 13 4

−1 5

, C =

−3 21 −4

6 2

, D = 2 3 1

1 −2 −3

.

Find the following matrices if they exist, or explain why they don’t exist. ( I stands for anidentity matrix of the appropriate size).

a) 3A, b) −2B, c) A + B, d) B + C , e) A + 3I ,f) B + 3I , g) AB, h) BA, i) BC , j) CD,k) A2, l) B2, m) (BD)2.

2. [H] Suppose A and B are matrices such that both AB and BA are defined.

a) Show that AB and B A are both square matrices.

b) If A and B are square matrices such that AB = BA show that(A − B)(A + B) = A2 − B2.

c) Find two 2 × 2 matrices A, B for which (A − B)(A + B) = A2 − B2.d) Prove that (A + B)2 = A2 + B2 + 2AB if and only if AB = BA.

3. [H] Let A and B be matrices of the same size. By considering the general entries [A]ij , [B]ij,[A + B]ij and [B + A]ij , prove the commutative laws of addition, i.e. A + B = B + A.

4. [H] Prove Proposition ?? on page ??.

5. [H] Let A and B be two matrices such that AB is defined. By considering the general entryin both sides of the equation, show that A(λB) = λAB where λ is any real number.

6. [R] Let

A =

1 0 10 1 11 1 2

, B = 1 22 −2

−1 4

, C = 2 23 −2

−2 4

.Show that AB = AC and deduce that matrices cannot in general be cancelled fromproducts.

7. [R] Let

A =

2 13 −1

.

Show that A2 = A + 5I and hence find A6 as a linear combination of A and I .

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CHAPTER 3. MATRICES 25

8. [R] Let

N =0 1 0

0 0 10 0 0

.

Find N 2 and N 3. Show that (I + N ) (I − N + N 2) = I .

9. [H] Let A and B be n × n real matrices such that A2 = I , B2 = I and (AB)2 = I . Prove thatAB = BA.

10. [H] Let A be a 2 × 2 real matrix such that AX = X A for all 2 × 2 real matrices X. Show thatA = αI for some α ∈ R.

11. [H] Suppose

A =

1 −2 34 0 13 2 −1

, B = 7 0 32 −1 6

−1 0 5

.a) Write down a column vector v such that Av is the second column of A.

b) Write down a row vector v such that vB is the third row of B .

c) Write down a column vector v such that Av is the second column of AB.

d) Write down a row vector v such that vB is the first row of AB.

12. [H] Prove parts ??, ?? and ?? of Proposition ?? on page ??.

Problems 3.2

13. [R] Find the transposes of the following matrices:

A =

1 −2−3 0

4 5

, B =

2 −5 4 3−4 6 5 5

5 0 8 6

, C =

1 4 24 −3 6

2 6 7

.

14. [R] Let a = (1, 3, −2)T and b = (0, 4, 2)T . Evaluate all of the following expressions that makesense and find those which are equal:

ab, aT b, abT , aT bT , bT a, baT .

15. [R] Suppose that A is a square matrix.

a) Show that the matrix B = (A + AT ) is symmetric.

b) Show that the matrix C = AAT is symmetric.

c) A matrix M with the property that M T = −M is called a skew symmetric matrix.Show that D = (A − AT ) is a skew symmetric matrix.

d) [H] Can you show how to write any square matrix as the sum of a symmetric and askew symmetric matrix?

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16. [H] Show by constructing an example that, in general, AT A = AAT , even if A is square.

17. [H] Suppose there exists a real matrix G such that GGT =

λ 00 µ

where λ, µ ∈ R. Provethat λ and µ are non-negative. If λ = 45 and µ = 20 find an example of such a matrix Gwith integer entries.

18. [H] Show that a matrix A ∈ M mn is a symmetric matrix if and only if:i) A is square (i.e., m = n), and

ii) xT Ay = (Ax)T y for all x, y ∈ Rn.

Problems 3.3

19. [R] Find the inverses of those of the following 2 × 2 matrices that have inverses.

a)

2 71 4

, b)

−4 73 −5

, c)

6 123 6

, d)

8 93 4

, e)

0 11 7

.

20. [R] Use the matrix inversion algorithm of Section 3.3 to decide if the following matrices areinvertible, and find the inverses for those which are invertible.

A =

1 3 −20 −1 20 0 1

, B =

0 2 01 2 3

−1 4

−2

, C =

1 2 32 3 44 5 6

, D =

1 4 12 3 11

−7

−2

.

21. [H] Write down the inverse of each of the following matrices

a)

1 0 00 5 0

0 0 6

b)

0 1 00 0 3

−2 0 0

.

22. [R] Decide if the following matrices are invertible, and find the inverses for those that areinvertible.

A =

1 2 −1 −11 2 −2 −20 1 1 11 4 0 −1

, B = 1 1 0 1

3 3 1 51 0 2 40 −4 2 1

, C = 1 2 −2 −72 4 3 14−1 −2 3 113 5 2 12

.

23. [R] Given that A, B and C are invertible n × n matrices simplifya) A(CB2A)−1C , b) (ABA−1)6, c) A(A−1 + A)2A−1,

d) A(I + (I − A) + · · · + (I − A)m).HINT: Write the first A as I − (I − A).

24. [R] a) Simplify (B−1A)−1.

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b) Find (B−1A)−1 if A−1 =

1 2 10 −1 10 0 1

and B =

1 0 10 2 01 0 3

.

25. [H] a) Prove that (AT )−1 = (A−1)T for any invertible matrix A.b) If A, B , C are invertible matrices of the same size simplify

i) A−1(BAT )T B, ii) AT (CAT )−1C T .

26. [R] Let A =

1 −1 11 1 0

3 −2 2

.

a) Calculate A−1. b) Solve Ax = c for x, where c = c1c2c3.

27. [H] A square matrix Q is said to be an orthogonal matrix if it has the property that QT Q = I .That is, QT = Q−1. Show that the matrix

Q =

23

13 − 23

23 − 23 1313

23

23

is orthogonal. Hence write down the solution of Qx = b for b ∈ R3.

28. [H] Show Q =

cos θ − sin θsin θ cos θ

is orthogonal. Show that x ∈ R2 and Qx are equidistant

from the origin. Show also that Q acts as a rotation on R2.

29. [X] A complex generalisation of Question 27 is the following. A square matrix Q is said to

be a unitary matrix if it has the property that QT

Q = I , where Q is the matrix obtainedfrom Q by taking complex conjugates of each entry of Q. Show that a unitary matrix Q

satisfies QT

= Q−1.

30. [X] Show that the matrix

Q =

1√ 2

i

− 1√ 2

i 01√ 2

i 1√ 2

i 0

0 0 −1

is unitary. Then use the result of Question 29 to write down the solution of Qx = b, whereb = (b1 b2 b3)

T with b1, b2, b3 complex.

31. [H] a) Suppose ab = 0. Write down the inverse of

a 0c b

.

b) Let A, B, C be 2 × 2 matrices where A and B are invertible and let O be the 2 × 2zero matrix. Find the inverse of the 4 × 4 matrix

A OC B.

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Problems 3.4

32. [R

] Evaluate the determinants of the following 2 × 2 matrices and hence determine whetheror not they are invertible.

a)

2 71 4

, b)

−4 73 −5

, c)

5 210 4

, d)

8 93 4

, e)

11 1312 14

.

33. [R] Evaluate the determinants for the following matrices by reducing to row echelon form.

a)

−1 1 22 4 −1

0 −1 1

, b)

1 −2 43 1 −2

1 5 −10

, c)

1 0 43 1 −2

1 5 −10

.

34. [R] Find the determinant of the matrix

1 1 1 11 1 7 11 8 3 11 1 1 4

.

35. [H] Suppose A =

a b cd e f

g h i

has determinant 5. Find

a) det

3a 3b 3c2d 2e 2f −g −h −i

, b) det

a + 2d b + 2e c + 2f d − g e − h f − i

g h i

,

c) det

d e f g h ia b c

, d) det(7A).36. [R] Given that A is a 3 × 3 matrix with det A = −2. Calculate:

a) det AT , b) det A−1, c) det A5.

37. [R] Evaluate det(A), det(B) and hence det(AB), where

A =

1 −2 30 3 5

3 4 −2

, B =

5 −1 0−3 2 4

2 5 0

.

38. [R] For what values of a is the matrix

1 2 21 3 1

1 3 a

invertible?

39. [H] Long long ago, a mathematician wrote C and C −1 on a piece of paper. Unfortunatelyinsects have damaged the paper and all that is left is

C =

−2 −1 1

1 2 −1∗ ∗ ∗

and C −1 =

∗ 0 −12 ∗ −15 1 ∗

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a) Find C −1. b) Find C. c) Find det C.

40. [H] Show that

det

1 1 1 11 1 + a 1 11 1 1 + b 11 1 1 1 + c

= abc.

41. [H] Let U 1 and U 2 be two n×n row-echelon matrices. Prove that det(U 1)det(U 2) = det(U 1U 2).

42. [H] Let A =

α 1 −1α 2α + 2 α

α

−3 α

−3 α

−3

.

a) Factorise det(A).

b) Hence, find the values of α will there be a nonzero solution of Ax = 0.

43. [R] Show by constructing an example that in general det(A + B) = det(A) + det(B).

44. [R] Show by constructing an example that in general det(λA) = λ det(A).

45. [H] Use the product rule for determinants to show that a square orthogonal matrix Q (seeQuestion 27) has a value for det(Q) of +1 or −1.

46. [X] Use the product rule for determinants to show that a square unitary matrix Q (see Ques-tion 29) has det(Q) = eiθ for some angle θ .

47. [H] Let A and B be two matrices which differ only in the first column, i.e., let

A =

a1 a2 · · · an

and B =

b1 a2 · · · an

,

where a1 and b1 are the first columns of A and B and where ai, i = 2, 3, . . . , n , are theremaining columns of both A and B. Let C be the matrix

C =

a1 + b1 a2 · · · an

obtained by replacing the first column of A (or B) by the sum of the first columns of Aand B .

Show thatdet(C ) = det(A) + det(B).

Explain why the result of this question also holds for adding two matrices which differonly in one column (not necessarily the first) or which differ only in one row.

48. [H] Show that

det

1 a a21 b b2

1 c c2

= (a − b)(b − c)(c − a).

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49. [H] Show that

det

1 + x 2 3 4

1 2 + x 3 41 2 3 + x 41 2 3 4 + x

= x3(x + 10).

50. [H] Factorise the determinant

det

y + z + 2x y zx z + x + 2y z

x y x + y + 2z

.

51. [H] Factorise the determinant

det

z 1 21 z 3

1 1 z + 1

and hence solve the simultaneous equations

zx + y = 2, x + zy = 3, x + y = z + 1.

52. [H] Suppose α, β and γ are the roots of the cubic equation x3+ px+q = 0 and sk = αk+β k+γ k.

Find s1

, s2

, s3

in terms of p and q and show that

det

s1 s2 s3s2 s3 s1

s3 s1 s2

= 8 p3 + 27q 3.

53. [H] Let A(x1, y1), B(x2, y2), C (x3, y3) be three points in the plane.

a) Suppose A, B and C are collinear. Show that

detx1 y1 1x2 y2 1

x3 y3 1

= 0.

b) Now suppose that A, B,C are not collinear. By considering the areas of some trapezia(or otherwise), show that the area of the triangle with vertices A,B, C is given by|D| where

2D = det

x1 y1 1x2 y2 1

x3 y3 1

.

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Problems 3.5

54. [R] A certain species of animal lives for at most 3 years. 50% of 0 – 1 year olds survive to be1 – 2 years old, 25% of 1 – 2 year olds survive to be 2 – 3 years old and then they all die.Each female 1 – 2 year old produces 4 female offspring in a year and each 2 – 3 year oldfemale produces 3 female offspring in a year.

a) If there are 1000 females in each age group initially, how many in each age group arethere after 1, 2, 3, 4, 5 and 10 years.

b) The total number of animals is clearly increasing. Divide the state vector for year kby the total number of animals in year k to get a vector of the proportions of eachage group in that year. Investigate the proportions as k → ∞ and the yearly increasein total number of animals as k → ∞.

55. [R] A car rental agency has 3 rental locations 1, 2 and 3. Cars may be rented at any locationand returned at any location. It has been found that the proportion of cars rented fromlocation j and returned to location i s a constant aij given by the matrix

A =

0.8 0.3 0.20.1 0.2 0.6

0.1 0.5 0.2

a) If a car is rented from location 2, what are the probabilities it will be at each locationafter 1, 2, 3, 4, 5, 10 days?

b) What are the probabilities it will be at each location after k days as k → ∞? Exper-iment with the car starting at location 1 or location 2.

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32 CHAPTER 3. MATRICES


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Chapter 4

VECTOR GEOMETRY

4.1 Lengths

4.2 The dot product

4.3 Applications: orthogonality and projection

4.4 The cross product

4.5 Scalar triple product and volume

4.6 Planes in R3

4.7 Projections and least squares approximations

4.8 Vector Geometry and MATLAB

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Problems 4.2

1. [R] Find the angles between the following pairs of vectors:

a)

−22

0

,

03

0

; b)

10

3

,

−25

1

; c)

71

−2

,

3−11

5

; d)

3014

,

−2613

.

2. [H] Find the cosines of the internal angles of the triangles whose vertices have the following

coordinate vectors:

a) A

40

2

, B

62

1

and C

51

6

; b) A

02

1

, B

−13

0

and C

31

2

;

c) A

1−2

03

, B

04

−25

and C

−2103

.

3. [R] A cube has vertices at the 8 points O (0, 0, 0), A (1, 0, 0), B (1, 1, 0), C (0, 1, 0), D (0, 0, 1),E (1, 0, 1), F (1, 1, 1), G (0, 1, 1). Sketch the cube, and then find the angle between the

diagonals −−→OF and

−→AG.

4. [H] Prove the following properties of dot products for vectors a, b, c ∈ R3. :a) a · b = b · a, b) a · (λb) = λ(a · b), c) a · (b + c) = a · b + a · c.

5. [H] Prove that |a| − |b| |a − b| for all a, b ∈ Rn.

HINT. See the proof of Minkowski’s inequality in Section ??.

6. [H] Use the dot product to prove that the diagonals of a square intersect at right angles.

Problems 4.3

7. [R] Let u1 =

1√ 2

0− 1√

2

, u2 =

01

0

, u3 =

1√ 2

01√ 2

and a =

2−3

1

. Show that the set of

vectors {u1, u2, u3} is an orthonormal set. Find scalars λ1, λ2, λ3 such that a = λ1u1 +λ2u2 + λ3u3.

HINT. See Examples ?? of Section 4.3.

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CHAPTER 4. VECTOR GEOMETRY 35

8. [H] Consider the triangle ABC in R3 formed by the points A(3, 2, 1), B(4, 4, 2) and C (6, 1, 0).

a) Find the coordinates of the midpoint M of the side B C.

b) Find the angle BAC.

c) Find the area of the triangle ABC.

d) Find the coordinates of the point D on B C such that AD is perpendicular to BC.

9. [R] Find the following projections:

a) the projection of

21

4

on

1−2

1

,

b) the projection of

2

−124

on −1

302

,

c) the projection of

−22

7

on the direction of the line x =

10

2

+ λ

−11

2

.

10. [R] Find the shortest distances between

a) the point (−2, 1, 5) and the line x =

12

−5

+ λ

63

−4

;

b) the point (0, 3, 8) and the line x1 − 1

1 =

x2 − 2−1 =

x3 − 34

;

c) the point (11, 2, −1) and the line of intersection of the planes

x · 1−1

3

= 0 and x = λ1

21

2

+ λ2

31

−3

.

11. [H] A point P in Rn has coordinate vector p. Find the coordinate vector of the point Q whichis the reflection of P in the line ℓ which passes through the point a parallel to the directiond.

NOTE. Define Q to be the point which lies in the same plane as P and ℓ with ℓ bisectingthe interval P Q.

12. [H] Let Q be a square n × n orthogonal matrix, i.e., a square matrix for which QT Q = I ,where I is an identity matrix. Show that (Qx) · (Qy) = x · y for all x, y ∈ Rn.HINT. x · y = xT y.

13. [H] Let Q be a square n × n orthogonal matrix. Show that the columns of Q are a set of northonormal vectors in Rn. Show that the rows of Q also form a set of n orthonormalvectors in Rn.

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14. [H] Let Q be a square n × n orthogonal matrix. Let {e1, e2, . . . , en} be the n standard basiscolumn vectors of Rn. Show that the set of vectors {Qe1, Qe2, . . . , Qen} also form a set of orthonormal vectors.

15. [H] Fix a, b ∈ Rn with b = 0. Let q (λ) = |a − λb|2.

a) Show q (λ) is a minimum when λ = λ0 = a · b|b|2 .

b) Determine q (λ0) and hence show that −|a| |b| a · b |a| |b|.

16. [H] Let B be a point in Rn with coordinate vector b. Let x = a + λd, λ ∈ R be the equationof a line. Do the following:

a) Show that the square of the distance from B to an arbitrary point x on the line is

given byq (λ) = |b − a|2 − 2λ(b − a) · d + λ2|d|2.

b) Find the shortest distance between the point B and the line by minimising q (λ).

c) If P is the point on the line closest to B , show that

−−→P B = b − a − projd (b − a),

and show that −−→P B is orthogonal to the direction d of the line.

NOTE. This problem proves that the shortest distance between a point and a line isobtained by “dropping a perpendicular from the point to the line”.

Problems 4.4

17. [R] Find the cross product a × b of the following pairs of vectors:

a) a =

02

−4

and b =

13

2

, b) a =

31

4

and b =

−26

1

,

c) a =

19

2

and b =

20

−5

.

18. [R] Find a vector which is perpendicular to

13

2

and

−20

4

.

19. [H] Prove the following properties of cross products for vectors a, b, c ∈ R3:a) a × a = 0; b) a × b = −b × a;c) a × (λb) = λ(a × b); d) a × (b + c) = a × b + a × c.

20. [R] Find the areas of, and the normals to the planes of, the following parallelograms:

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a) the parallelogram spanned by

13

2

and

02

4

;

b) a parallelogram which has vertices at the three points A (0, 2, 1), B (−1, 3, 0) andC (3, 1, 2) and sides

−→AB and

−→AC .

21. [R] Find the areas of the triangles with the following vertices:

a) A (0, 2, 1), B (−1, 3, 0) and C (3, 1, 2);b) A (2, 2, 0), B (−1, 0, 2) and C (0, 4, 3).

22. [R] Let D, E, F be the points with coordinate vectors

d =

567

, e = 678

, f = 7810

a) Calculate cos(∠DEF ) as a surd.

b) Calculate the area of ∆DEF as a surd.


a) the line through (1, 2, 3) parallel to

301

and the line through (0, 2, 5) parallel to

3−2

2

;

b) the line through the p oints (1, 3, 1) and (1, 5, −1) and the line through the points(0, 2, 1) and (1, 2, −3);

c) the lines x =

27

8

+ λ

43

−5

and x1 − 1−10 = x2 − 21 = x3 − 34 .

24. [H] Let a, b, c be three vectors in R3 which satisfy the relations b = c

×a and c = a

×b.

a) Show that a, b and c are a set of mutually orthogonal vectors.

b) Show that b and c are of equal length and that if b = 0, then a is a unit vector (i.e.a vector of length 1).

25. [H] A tetrahedron has vertices A, B, C and D with coordinate vectors for the points being

a =

01

2

, b =

−14

1

, c =

10

3

and d =

−31

2

. Find parametric vector equations

for the two altitudes of the tetrahedron which pass through the vertices A and B, anddetermine whether the two altitudes intersect or not.

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NOTE. An altitude of a tetrahedron through a vertex is a line through the vertex andperpendicular to the opposite face.

26. [H] Points A, B, C and D have coordinate vectors

10

2

,

−12

3

,

01

1

and

20

−1

, respec-

tively.

a) Find a parametric vector equations of the line through A and B and the line throughC and D .

b) Find the shortest distance between the lines AB and C D.

c) Find the point P on AB and point Q on CD such that P Q is the shortest distancebetween the lines AB and C D.

Problems 4.5

27. [R] Prove that

a · (b × c) = deta1 a2 a3b1 b2 b3

c1 c2 c3

.

28. [R] Find the volumes of the following parallelepipeds:

a) the parallelepiped spanned by

21

3

,

41

2

and

02

1

;

b) a parallelepiped which has vertices at the four points A (2, 1, 3), B (−2, 1, 4), C (0, 4, 1)and D (3, −1, 0), with sides

−→AB,

−→AC and

−→AD.

29. [R] Show that the four points A,B,C,O with coordinate vectors

21

3

,

41

2

,

61

1

,

00

0

are coplanar.

30. [H] Prove the following relationships between “volumes” and determinants:

a) In two dimensions, let a =

a1a2

and b =

b1b2

, and consider the parallelogram

spanned by a and b. Show that a parametric vector form for the parallelogram is

x = λ1a + λ2b for 0 λ1 1, 0 λ2 1,

and then show that the area of the parallelogram is equal to |det(A)|, where A is thematrix with rows a and b.

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b) In three dimensions, let a =

a1a2

a3

, b =

b1b2

b3

and c =

c1c2

c3

, and consider the

parallelepiped spanned by a, b and c. A parametric vector form for the equation of the parallelepiped is

x = λ1a + λ2b + λ3c for 0 λ1 1, 0 λ2 1, 0 λ3 1.

Show that the volume of the parallelepiped is equal to | det(A)|, where A is the matrixwith rows a, b and c.

c) What is the one-dimensional version of these results?

Problems 4.6

31. [R] Find parametric vector, point-normal, and Cartesian forms for the following planes:

a) the plane through (1, 2, −2) perpendicular to−11

2

;

b) the plane through (1, 2, −2) parallel to−11

2

and

23

1

;

c) the plane through the three points (1, 2, −2), (−1, 1, 2) and (2, 3, 1);

d) the plane with intercepts −1, 2 and −4 on the x1, x2 and x3 axes;e) the plane through (1, 2, −2) which is parallel to

−12

−2

and the line of intersection

of the planes

x ·12

3

= 0 and x = λ1

21

2

+ λ2

10

−1

.

32. [R] Consider four points O,A, B,C in R3 with coordinate vectors

0 =

00

0

, a =

12

4

, b =

10

−1

, c =

2−1

−1

.

Let Π be the plane through A and parallel to the lines OB and OC.

a) Find a parametric vector form for Π.

b) Show that Π passes through the point (12, −2, −3).c) Find a vector n normal to Π.

d) Use the point normal form to find a Cartesian equation for Π.

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33. [R] Consider the 3 planes Π1, Π2, Π3 with Cartesian equations

x + 3y + 2z = 5

2x + y − z = 27x + 11y + 4z = 13

a) Regard the 3 equations as a system of equations Ax = b and find the solution(s), if any, of the system.

b) What is det A?

c) Show that no pair of the planes are parallel.

d) Using a) and c), describe the configuration of the planes Π1, Π2 and Π3 in R3.

34. [R] Find the following projections:

a) the projection of

23

8

on the normal to the plane 2x1 + 2x2 + x3 = 4;

b) the projection of

22

−1

on the line of intersection of the planes

x ·1

−13

= 0 and x = λ1

212

+ λ2

31−3

.


a) the point (2, 6, −5) and the planex −

12

3

·

−24

4

= 0;

b) the point (1, 4, 1) and the plane 2x1 − x2 + x3 = 5;c) the point (1, 2, 1) and the plane with intercepts at 3, −1, 2 on the three axes;d) the origin and the plane through the three points (2, 1, 3), (5, 3, 1) and (5, 1, 2).

36. [R] Let P be the plane in R3 through the points A = (1, 2, 0), B = (0, 1, 2), and C = (−1, 3, 1).a) Find a parametric vector form for the plane P .

b) Find a vector n normal to the plane P .

c) Find a point normal form for the plane P .

d) Find the shortest distance from the point Q = (2, 4, 5) to the plane P.

37. [H] a) Let a and v be two non-zero vectors in R3. Show how to write v as c + d where c isparallel to a and d is perpendicular to a.

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b) Consider the plane

Π =xy

z

: xyz

= 210

+ λ1121

+ λ2 01−1

, λ1, λ2 ∈ R

and the vector v =

11

1

. By using a) (or otherwise) express v as c + d where d is

parallel to Π and c is perpendicular to Π. (We call d the projection of v onto Π).

Problems 4.7

38. [R] Suppose v1 =

0210

, v2 =

1−100

, v3 =

120

−1

, v4 =

1001

.

a) Find projπv3 where π = span(v1, v2)

b) Find projσv4 where σ = span(v1, v2, v3).

39. [H] Prove that |Ax − y| is least when y − Ax is orthogonal to σ = span(columns of A).

40. [H] Given a set of vectors

{v1, v2, . . . , vn

}such that none of the vectors is a linear combination

of ones earlier in the list, prove that the following procedure constructs an orthonormalset {u1, u2, . . . , un} with the property that for each k, 1 k n,span{u1, . . . , uk} = span{v1, . . . , vk} = σk.

Set u1 = v1

|v1| .For k = 2, . . . , n in turn set

wk = vk − projσk−1vk,uk =

wk

|wk| .

41. [R] Use the procedure of problem 40 to find an orthonormal spanning set for the vectors of problem 38 and hence find projπv3 and projσv4.

42. [R] Given the points (ti, yi), i = 1, . . . , 5 : (−1, −14), (0, −5), (1, −4), (2, 1), (3, 22).

a) Find the straight line y = α + βt of best fit to this data.

b) Set up the normal equations for the cubic y = α + βt + γt2 + δt3 of best fit to thisdata.

c) [H]Solve the equations in b).Note: they are rigged to come out nicely.

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43. [R] A dependent variable z depends on two independent variables x, y by a relationship

z = α + βx + γy + δxy.

If z is observed for n sets of x, y values giving data

(x1, y1, z1), . . . , (xn, yn, zn)

derive the normal equations for the surface of best fit to this data.

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Chapter 5

INTRODUCTION TOPROBABILITY AND STATISTICS

5.1 Introduction

5.2 Some preliminary set theory

5.3 Probability

5.4 Discrete random variables

5.5 Continuous random variables

5.6 The Normal Distribution

5.7 Sums and means of random variables

5.8 Approximations to the Binomial Distribution

5.9 Probability and MATLAB

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Problems 5.1

1. [R] (For discussion) Analyse the following claims.

a) Eighty five percent of lung cancer sufferers have been smokers. Therefore, if we bansmoking we will reduce the lung cancer deaths by 85%.

b) A white defendant was identified by a black witness. The defendant’s lawyer claimsthat studies show that people make 3 times as many mistakes identifying people of other races as they make identifying people of their own race. The black witness’s

evidence is therefore not to be trusted.

c) A certain town has 4 times the expected number of leukaemia sufferers. There musttherefore be an environmental reason for these extra cases of leukaemia.

Problems 5.2

2. [R] Let A = {a, c, d, e} and B = {d, e, f }.Suppose that the universal set is S = {a, b, c, d, e, f }.Write down the following sets.

a) A − B, b) B − A, c) A ∪ Ac

, d) B ∩ Bc

,e) A ∪ Bc, f) Ac ∩ B, g) (A ∪ B)c, h) Ac ∩ Bc.

3. [R] Find all the subsets of {∅, {∅}, {{∅}}}.

4. [R] Of the students taking ACTL1001 and MATH1151 in a hypothetical year, 90% passedMATH1151, 85% passed ACTL1001 and 6% passed neither. What percentage passedboth? What percentage of those who passed ACTL1001 also passed MATH1151?

5. [R] A brewery brews one type of beer which is marketed under three different brands. In asurvey of 150 first year students, 58 drink at least brand A, 49 drink at least brand B and

57 drink at least brand C. 14 drink brand A and brand C, 13 drink brand A and brandB and 17 drink both brand B and brand C. 4 students drink all three brands. How manystudents drink none of these three brands?

6. [R] A company surveys 90 UNSW students to find their reasons for choosing which brand of mobile phone to purchase. The survey findings are that 36 students make their decisionon the basis of the available ring tones, 45 students make their decisions based on the sizeof the phone and 37 are influenced by the available cases. 28 students are influenced bythe available ring tones and the size of the phone, 18 by the available ring tones and casesand 23 by the available cases and the size of phone. 7 students take all three issues intoaccount. Are the results of this survey valid?

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CHAPTER 5. PROBABILITY AND STATISTICS 45

7. [R] Suppose A, B and C represent three events. Using unions, intersections and complements,find expressions representing the events

a) only A occurs,

b) at least one event occurs,

c) at least two events occur,

d) exactly one event occurs,

e) exactly two events occur.

Problems 5.3

8. [R] A survey was carried out in a new development area to gain data on home-delivered

newspapers. 110 homes were selected at random and the occupants asked whether theyhad the daily paper or the weekend paper home delivered. 74 received the daily paper,58 received the weekend paper and 10 received no paper at all. Determine the probabilitythat the last home visited in this survey received both the daily and weekend papers.

9. [R] Two fair dice are thrown.

a) What is the probability that the sum of the two numbers obtained is 6?

b) What is the probability that both dice show the same number?

c) What is the probability that at least one of the dice shows an even number?

10. [R] What is the probability that a randomly chosen two digit positive integer is divisible by 3or 5?

11. [R] A system has n independent components and each fail with probability p. Calculate theprobability that the system will fail when

a) the components are in series, so the system fails if any one of the components fail;

b) the components are in parallel, so the system fails only when all of the componentsfail.

12. [R] The following is a table of the annual promotion probabilities at a particular workplace,

broken down by gender.

Promoted Not promoted Total

Male 0.17 0.68 0.85Female 0.03 0.12 0.15

Is there gender bias in promotion?

13. [R] Urn 1 contains 2 red balls and 3 black balls. Urn 2 contains 4 red balls and 5 black balls.

a) If an urn is randomly selected and a ball drawn at random from it, what is theprobability that the ball is red?

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b) Suppose a ball is drawn at random from urn 1 and placed into urn 2. If a ball is thendrawn at random from the 10 balls in urn 2, what is the probability that it is red?

c) In the previous part, given that the ball drawn from urn 2 is red, what is the proba-bility that the ball transferred from urn 1 was black?

14. [R] Employment data at a large company reveal that 72% of the workers are married, that44% are university graduates and that half of the university graduates are married. Whatis the probability that a randomly chosen worker

a) is neither married nor a university graduate?

b) is married but not a university graduate?

c) is married or is a university graduate?

15. [R] Suppose that 30% of computer users use a Macintosh, 50% use a Microsoft Windows PCand that 20% use Linux. Also suppose that 60% of the Macintosh users have succumbed toa computer virus, 80% of the Windows PC users get the virus and 10% of the Linux usersget the virus. A computer user is selected at random and it is found that her computerwas infected with the virus. What is the probability that she is a Windows PC user?

16. [R] Down’s syndrome is a disorder that affects 1 in 270 babies born to mothers aged 35 or over.A new blood test for the condition has a sensitivity (i.e. the probability of a positive testresult given the Down’s syndrome is present) of 89%. The specificity ( i.e. the probabilityof a negative test result given that Down’s syndrome is absent) of the new test is 75%.

a) What proportion of women over age 35 would test positive on this new blood test?

b) A mother over age 35 receives a positive test result. What is the chance that Down’ssyndrome is actually present?

c) A mother over age 35 receives a negative test result. What is the chance that Down’ssyndrome is actually present?

17. [H] Tom and Bob play a game by each tossing a fair coin. The game consists of tossing thetwo coins together, until for the first time either two heads appear when Tom wins, or twotails appear when Bob wins.

a) Show that the probability that Tom wins at or before the nth toss is 12 − 1

2n+1.

b) Show that the probability that the game is decided at or before the nth toss is 1− 12n

.

18. [R] On the basis of the health records of a particular group of people, an insurance companyaccepted 60% of the group for a 10 year life policy. Ten years later it examined the survivalrates for the whole group and found that 80% of those accepted for the policy had survivedthe 10 years, while 50% of those rejected had survived the 10 years. What percentage of the group did not survive 10 years? If a person did survive 10 years, what is the probabilitythat they had been refused cover?

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CHAPTER 5. PROBABILITY AND STATISTICS 47

19.

algebra probs

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