lesson 1 - linear system,matrices,determinants

7/30/2019 Lesson 1 - Linear System,Matrices,Determinants

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available at: www.groups.yahoo.com/mapuam15


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Is a set of equations containing two or moreunknowns having similar solution set.

Linear SystemsIs a set oflinear equations containing two or

more unknowns having similar solution set.


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Homework 1: Solve each of the given systemsand determine the number of solutions.

1. 2.

3. 4.

943 yx

1087 zyx

0852 yx

254 zxx3825 xyx

5x 3y 2 0

626 zyx

7x 3y 7

03742 zyx

10 zyx


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1. Independent or consistent system

System of equations in two or more variables represented by curves

intersecting at a common point.

Have finite numberof solutions represented by the points of

intersection of the curves.2. Inconsistent system

System of equations in two or more variables represented by non-

intersecting curves.

System with no solution

3. Dependent system

System of equations in two or more variables represented by curves

coincident with one another.

System with infinite numberof solutions.


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Geometrically, the solution to the linear system leads to three

possibilities:The system has a unique solution; that is, the two lines

intersect at exactly one point.

The system has no solution; that is, the lines do not intersect.

The system has infinitely many solutions; that is, the linescoincide.

x

y

x

y

x

y


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1. 2.

3. 4.

5.

8yx

15zy3x4

1yx2

2z2yx 422 zyx

62 yx

9y6x8

1)zy(3

1x

5x3y

1)2(21 xzy

10y2x6

2)yx2(4

1z


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Graphical Method of Solving a System of Linear EquationsThe graphical method of solving a system of linear equations is a method

that determines the solutionin terms of the common point(s)or the

point(s) of intersectionamong the graphs representing each of the

equations in the system.

The following are the basic steps to be followed:

1. Draw the graphs associated to the equations of the system.

2. Determine the common point or the point of intersection among the graphs.3. Read the coordinates of the point giving the solution ( x , y ).


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Find the solution to each of the given system by the graphical method.

1. 2.

1. S (3, 1) 2. S (1, 2)

9y3x2 13y4x3

X Y1 Y20 3 13/4

1 7/3 5/2

2 5/3 7/4

3 1 1

EQ1

EQ26

11

2

y

3

x

4

3

3

y

4

x

EQ1 EQ2

X Y1 Y2

0 11/3 9/4

1 3 3

2 7/3 15/4

3 5/3 9/2


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Analytical MethodsElimination Of a Variable by Addition/Subtraction

This is an analytical method of solving a system of

equations that eliminates a variable addition/subtraction of

multiple equations.

Elimination Of a Variable by Substitution

This is an analytical method of solving a system of

equations that eliminates a variable by replacing one of the

variables in one of the equations by an equal expressions

obtained from the other equation.


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3. An oil refinery produces low-sulfur and high sulfur fuel. Each

ton of low-sulfur fuel requires 5 minutes in the blending plant

and 4 minutes in the refining plant; each ton of high-sulfurfuel requires 4 minutes in the blending plant and 2 minutes in

the refining plant. If the blending plant is available for 3

hours and the refining plant is available for 2 hours, how

many tons of each type of fuel should be manufactured so thatthe plants are fully utilized?

4. A dietician is preparing a meal consisting of foods A, B, and

C. Each ounce of food A contains 2 units of protein, 3 units of

fat, and 4 units of carbohydrate. Each ounce of food Bcontains 3 units of protein, 2 units of fat, and 1 unit of

carbohydrate. Each ounce of food C contains 3 units of

protein, 3 units of fat, and 2 units of carbohydrate. If the


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meal must provide exactly 25 units of protein, 24 units of fat, and

21 units of carbohydrate, how many ounces of each type of foodshould be used?


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At the end of the lesson, the student should be

able to:

Define matrix

Identify different types of matrices.

Perform operations on matrices.

Define determinant of matrix.

Evaluate determinant of a square matrix.


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Matrix is a rectangular array of elements

arranged in mrows and ncolumns, and is

enclosed by a pair of parenthesis ( ), braces [ ]

or brackets { }. The elements maybe

numbers, variables.


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Matrix can be written in the general form:

mna...m2am1a

............2n

a...22

a21

a1na...12a11a

)ij

(aA

row column

Uppercase letterLower case letter Listed elements


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1. Column Vector or Column Matrix

0

8

4

3

A

is a matrix with only one column and m rows

(mx1).


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2. Row Vector or Row Matrix

6402A

is a matrix with only one row and ncolumns

(1xn).


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3. Square Matrix

422

303

012

A

is a matrix with equal number ofrows and

columns.


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4. Symmetric Matrix

423

201

312

A

is a matrix with ai j

= aji

.


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5. Diagonal Matrix

400

070

002

A

is a square matrix whose elements above and

below the main diagonal is zero.


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6. Identity Matrix or Unit Matrix

100

010

001

A

is a diagonal matrix with all elements on the

main diagonal equal to 1.


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7. Upper Triangular Matrix

800

320

236

A

is a matrix with all elements below the main

diagonal equal to 0.


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8. Lower Triangular Matrix

834

027

006

A

is a matrix with all elements above the main

diagonal equal to 0.


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9. Zero One Matrix

011

00

11

0

0

A

is a matrix consisting ofzeros and ones only

as entries.


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10. Transpose of a Matrix

011

0011

0

0

A

The transpose of a matrix , denoted by AT , is

obtained by interchanging the rows and

columns of the matrix.

000

101101TA


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A. Matrix Equality

ijbaij BA

Two matrices of the same dimension (mxn)

are equal if and only if all elements of the first

matrix is equal to its respective element in the

second matrix

for all is and js


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Find the values of a, b, c, and d so that

each of the given statements would be true:

a. b.

01

2

031

522

c

ba

2

7

2

1

24

47

20

01

b

d

a

c

=


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B. Matrix Addition

ijbac ijij

BAC

The sum of matrices of the same dimension

(mxn) is the sum ofall elements of the same

position. for all is and js

44

92

0422

4531

02

43

42

51


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C. Scalar Multiplication

84

102

42

512

)( ijakAk

The scalar product of matrix is obtained by

multiplying a constant k to every element of

the matrix

for all is and js


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Properties of Matrix Addition and Scalar Multiplication

If A, B, C and O (zero matrix) are m x n matrices and c and d are scalar

numbers, then the following hold true.

1. A + B = B + A Commutative Property of Addition

2. A + (B + C) = (A + B) + C Associative Property of Addition

3. cdA = c(dA) Associative Property of Scalar

Multiplication

4. IA = A Scalar Identity

5. c(A + B) = cA + cB Distributive Property6. (c +d) A = cA + dA Distributive Property

7. A + O = A Identity Property of Addition

Note: The difference AB = A + (-B)


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Vector Product

Multiplying a row vector (1 x n) by a column

vector (n x 1) results to a 1 x 1 matrix

nn

n

nbabababa

b

b

bb

aaaa

......3322113

2

1

321


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C. Matrix Multiplication

A vector product of matrices A and B can be

obtained if the number ofcolumns of the

multiplicand is equal to the number ofrows

of the multiplier

nkkmnmBAC


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The product of an m x k matrix A and a

k x n matrix B is an m x n matrix C whose

entry in the ith row and jth column is the

vector product of the ith row of A and the

jth column of B

nkkmnmBAC


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nkkmnm BAC

ijcC = (ith row of A)(jth column of B)


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C. Matrix Multiplication or Vector Product

p

kkjikij bac 1

7120

5712

1127

130

211

307

013

322

401

x


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p

kkjikij bac 1

7120

5712

1127

130

211

307

013

322

401

x


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p

kkjikij bac 1

7120

5712

1127

130

211

307

013

322

401

x


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Example: Given

2 -33 0 -4

A = and B = 4 -1-2 2 -1

1 5


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Given the following matrices, perform

the indicated operations

a. A *B -2A b. B*AT + 2( B)

97

83

52

31BA


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Example: Given

2 -33 0 -4

A = and B = 4 -1-2 2 -1

1 5

Find the following:1. AB

2. (2B)(3A)


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)det(AA

A scalar quantity associated with a square

matrix.


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12

53A

The determinant of an order 2 matrix (2 x 2 matrix) can be

obtained by taking the difference of the products of the

diagonal elements.

40

21B

|A| = det(A) = 3*1 -2*5

|A| = -7|B| = det(B) = -1*4 -0*2

|A| = -4


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12

03B

Evaluate the determinants of the following matrices:

1. 4.

2. 5.

3. 6.

03

12A

41

32C

18

24D

21

53E

12

510F


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For a 3x3 matrix, the determinant is:

333231

232221

131211

||aaaaaa

aaa

A

)()()(|| 312232121333213123123223332211 aaaaaaaaaaaaaaaA


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The determinant of an order 3 matrix (3 x 3 matrix) can beobtained by doing the following steps:

1.Copy the first two columns of the given matrix as the 4th

and 5th columns.

2.Multiply the downward diagonal elements of theresulting matrix. Find the sum of these three products.

3.Multiply the upward diagonal elements of the matrix.

Find the sum of these three products.

4.Subtract the result of (2) by the result of (3). This is the

determinant.


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221

103

212

A

Evaluate the following determinant:

21

03

12

|A| = det(A) = (2*0*-2+-1*1*1+2*3*2)

(1*0*2+2*1*2+-2*3*-1)

|A| = 1


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The minor of a matrixmi jabout an element ai j is asubmatrix of A where the ith row and jth column has

been removed.

The cofactor of a Matrix about the element ai jis the

determinant of the minor mi j preceded by

(negative) if(i+j) is odd.


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For any square matrix of order n>=3, the determinant can beevaluated using:

a) Co-factor Expansion about a column:

for all i, for a given j

b) Co-factor Expansion about a row:

for all j, for a given i ,

where: aij is the element in the ith-jth position

Aij is the cofactor of aij.

m

i

ijijAaA1

||

n

j

ijijAaA1

||


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221

103

212

A

Evaluate the following determinant:

Column aij mij Sign Aij aij*Ai j

1 3 - -(-2) 6

2 0 + +(-4) 0

3 1 - -(5) -5

TOTAL 1

Selecting row 2

21

22

22

21

21

12


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110

322

111

B

Evaluate the determinants of each of the followingmatrices, using both methods

1. 3.

2. 4.

204

121

211

A

111

422

301

D

414

123

031

F

lesson 1 - linear system,matrices,determinants

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