1

CHAPTER 1Matrix Algebra

2

INTRODUCTION

• A rectangular array of elements or entries aij involving m rows and n columns

• Matrix can be written as A = [aij ]

Matrix

11 12 13 1 1

21 22 23 2 2

31 32 33 3 3

1 2 3

1 2 3

. .

. .

. .

. . . . . . . [ ]

. .

. . . . . . .

. .

j n

j n

j n

ij m n

i i i ij in

m m m mj mn

n coloumns

a a a a a

a a a a a

a a a a a

A m rows a

a a a a a

a a a a a

3

Elements in matrix may be real or complex numbers, or even functions.

1 3 2 real number

2 9 1

2 3 4complex number

1 2

i i

i

cos sin function

sin cos

4

If i = j, then the elements is called the leading diagonal of matrix A.

Example 11 22 33, ,a a a

5

EXAMPLE

6

SOLUTION

i. Order of matrix : 3x3ii. Leading diagonal : 1,0,3iii. a12 = 2

a31= 5

a23= 2

7

TYPE OF MATRICES

8

Not diagonal

Not diagonal

Diagonal

9

Scalar

Scalar Not scalar

10Not identity matrix

Not identity matrix

11

2 4

2 7A

3 0

9 3

0 1

B

12

13

14

1 3 7

2 3 6

4 0 8

TA

9 1

3 2 4TB

15

T

5 1 3

1 2 2 since A =A, so it is a symmetric matrix

3 2 7

TA

16

17

T

0 1 5 0 1 5

1 0 3 , - 1 0 3

5 3 0 5 3 0

since A =-A, so it is a skew symmetric matrix

TA A

18

Row echelon form

Not Row echelon form

Row echelon form

Not Row echelon form

19

Reduced Row echelon form

Reduced Row echelon form

Not Reduced Row echelon form

Not Reduced Row echelon form

20

i) Not possible

1 2 3 7 8 9ii)

4 5 6 10 11 12

8 10 12

14 16 18

A B

A B

MATRIX OPERATION

21

i) Not possible 1 5 0

)10 1 2

ii B A

22

MATRIX OPERATION

23

24

2(3) 2(1) 2( 1)2

2(0) 2( 2) 2(4)

6 2 22

0 4 8

A

A

25

3(1) 1(2) 4( 6) 3( 1) 1(0) 4(4))

5(1) 2(2) ( 2)( 6) 5( 1) 2(0) ( 2)(4)

19 13

21 13

i AB

AB

26

27

3 2 3

3 0 1AB

Answer

28

DETERMINANT

29

)det

3(1) 6( 1)

9

i A A

)det

3 1

6 1

3(1) ( 1)(6)

=9

T T

T

T

ii A A

A

A

30

+++---

31

3( 5)(7) ( 1)(2)(0) 6(9)(4) 6( 5)(0) (2)(4)(3) ( 1)(9)(7)

111 39

150

A

A

A

32

33

11

211

11

2 2)

3 0

2(0) ( 2)( 3)

6

( 1) ( 6)

6

i M

C

C

34

35

11 11 12 12 13 13

2 3 45 2 9 2 9 5(3) ( 1) ( 1) ( 1) (6) ( 1)

4 7 0 7 0 4

129 63 216

150

A a c a c a c

A

A

A

36

37

i) Find all the value of Cij

ii) Arrange all the value in matrix form [Cij]

iii) Then transpose [Cij]

38

11 12 13

21 22 23

31 32 33

2, -6, 1

3, 0, 3

1, 3, 5

2 6 1 2 3 1

3 0 3 , then adj 6 0 3

1 3 5 1 3 5

T

c c c

c c c

c c c

Cij Cij

39

40

1

) 3(6) 4(2) 10

6 41

2 310

i A

A

41

42

11 12 13

21 22 23

31 32 33

1

0, 9, 18

2, 6, 11

1, 3, 1

0 9 18 0 2 1

2 6 11 , then adj 9 6 3

1 3 1 18 11 1

0 2 11

9 6 39

18 11 1

T

c c c

c c c

c c c

Cij Cij

A

43

44

45

EXERCISE:

Find Inverse of matrix A using elementary row operation

1 0 1

1 2 3

3 1 5

A

1

7 11

2 22 1 1

5 11

2 2

A

Answer

46

4 4 2

0 5 1

4 3 2

B

Find Inverse of matrix B using elementary row operation

1

7 1 3

4 2 21 0 1

5 1 5

B

Answer

47

SOLVING THE SYSTEMS OF LINEAR EQUATION

1.6.1 Inversion Method

1.6.2 Gaussian Elimination and Gauss-Jordan Elimination

1.6.3 Cramer’s Rule

48

1.6.1 INVERSION METHOD

AX = B

49

1 1A AX A B

1 1( )

A AX A A X

X

1X A B

The left-hand side can be simplified by noting that multiplying a matrix by its inverse gives the identity matrix

Multiplying a matrix by the identity matrix has no effect and so

50

1

7 2 12

3 1 5

1 21

3 77(1) 2(3)

1 2

3 7

AX B

x

y

A

1

,

1 2 12

3 7 5

1(12) ( 2)(5)

( 3)(12) 7(5)

2

1

, 2, 1

Then

X A B

x

y

hence x y

51

1 2 3

1 2 3

1 2 3

3 6

8 9 4 21

2 2 3

x x x

x x x

x x x

52

53

1.6.2 GAUSSIAN ELIMINATION AND GAUSS-JORDAN ELIMINATION

Gaussian elimination process.• The procedure to reduce matrix A to

row echelon form (REF) by using elementary row operation (ERO).

Gauss-Jordan elimination• The procedure to reduce matrix A to

reduced row echelon form (RREF) by using elementary row operation (ERO).

54

EXAMPLE 1.6.2i. Solve the system of linear equations by using

Gaussian elimination and Gauss-Jordan elimination.

1 2 3

1 2 3

1 2 3

2 2 8

3 3 4

4 2 1

x x x

x x x

x x x

55

GAUSSIAN ELIMINATION

2 1

3 1

23 2

3

22 1 4

1147

1

5

2 1 2 8 1 3 3 4

1 3 3 4 2 1 2 8

4 2 1 1 4 2 1 1

1 3 3 41 3 3 4

4 160 7 4 16 0 1

7 70 14 13 17

0 14 13 17

1 3 3 4 1 3

4 160 1

7 70 0 5 15

R RR R

R R R

R

R R

3 4

4 160 1

7 70 0 1 3

56

1 2 3

2 3

3

2

2

1

1

From the Gaussian elimintion we have

3 3 4

4 16

7 7 3

4 16(3)

7 74

3(4) 3(3) 4

1

x x x

x x

x

x

x

x

x

57

2 1

3 1

23 2

22 1 4

1147

From Gauss-Jordan Elimination,

2 1 2 8 1 3 3 4

1 3 3 4 2 1 2 8

4 2 1 1 4 2 1 1

1 3 3 41 3 3 4

4 160 7 4 16 0 1

7 70 14 13 17

0 14 13 17

1 3 3 4

4 160 1

7 70

R RR R

R R R

R R

31 2

135

1 3 3 4

4 160 1

7 70 5 15 0 0 1 3

R R R

58

1 3

2 3

9

74

7

1 2 3

9 201 0

7 7 1 0 0 14 16

0 1 0 1 0 47 7

0 0 1 30 0 1 3

1, 4, 3

R R

R R

x x x

59

60

1.6.3 CRAMER’S RULE

method of obtaining the solution of equations like these as the ratio of two determinants.

11211 byaxa 22221 byaxa

1 12

2 22

11 12

21 22

i

b aA

b ax

a aA

a a

11 1

12 2

11 12

21 22

i

a bA

a by

a aA

a a

61

62

63

1 3 1 8

2 1 1 7

1 1 1 2

,

8 3 1

7 1 1

2 1 1 162

1 3 1 8

2 1 1

1 1 1

x

y

z

then

x

1 8 1

2 7 1

1 2 1 12 31 3 1 8 2

2 1 1

1 1 1

y

1 3 8

2 1 7

1 1 2 12 31 3 1 8 2

2 1 1

1 1 1

z

64

1.7 MATRIX APPLICATION

1.7.1 Application of Inversion Method

1.7.2 Application of Gaussian Elimination and Gauss Jordan Elimination Method

1.7.3 Application of Cramer’s Rule

65

EXAMPLE 1.7.1.2 - APPLICATION OF INVERSION METHOD Grand Canyon Tours offers air and ground scenic tours of

the Grand Canyon. Tickets for the 7.5 hours tour cost RM169 for an adult and RM 129 for a child and each tour group is limited to 19 people. On three recent fully booked tours, total receipts were RM2931 for the first tour, RM3011 for the second tour and RM 2771 for the third tour. Determine how many adults and how many children were in each tour.

66

Let A= Adult

C= Children

From information above, the equation are

1

19

169 129 3011

Tour

A C

A C

1 1 19

169 129 3011

129 1 191

169 1 30111 129 1 169

129 1 191

169 1 301140

129 11940 40

169 1 3011

40 40

A

C

A

C

67

14

5

14,

5

A

C

68

EXAMPLE 1.7.2.1 – APPLICATION OF GAUSSIAN ELIMINATION AND GAUSS JORDAN ELIMINATION METHOD

Ahmad inherited RM25000 and invested part of it in a money market account, part in municipal bonds and part in a mutual fund. After one year, he received a total of RM1620 in simple interest from three investments. The money market paid 6% annually, the bonds paid 7% annually and the mutually fund paid 8% annually. There was RM6000 more invested in the bonds than the mutual funds. Find the amount Ahmad invested in each category using Gauss Elimination Method.

69

SolutionLet A= Money Market Account

B=Municipal bond C=Mutual fund

Arrange in matrix form

25000

0.06 0.07 0.08 1620

6000

A B C

A B C

B C

1 1 1 25000

0.06 0.07 0.08 1620

0 1 1 6000

70

2 1 2

3 23

1 1 1 25000 1 1 1 25000

0.06 0.07 0.08 1620 0.06 0 0.01 0.02 120 100

0 1 1 6000 0 1 1 6000

1 1 1 25000 1 1 1 25000

0 1 2 12000 0 1 2 12000

0 1 1 6000 0 0 3 6000 1

3

1 1 1

0 1 2

R R R

R R R

HHHHHHHHHHHHHH

HHHHHHHHHHHHHH

25000

12000

0 0 1 2000

71

25000

25000 8000 2000

15000

A B C

25000

2 12000

2000

A B C

B C

C

12000 2

12000 2(2000)

12000 4000

8000

B C

72

EXAMPLE 1.7.3.1-APPLICATION OF CRAMER’S RULE

A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission. Find out the rates of commission on the items A, B and C. Solve by Cramer’s rule.

Months Sales of Units Total commission drawnA B C

JanuaryFebruaryMarch

90 100 20 130 50 4060 100 30

800900850

73

SolutionChange into matrix form

800 100 20

900 50 40

850 100 30

90 100 20

130 50 40

60 100 30

50 40 900 40 900 50800 100 20

100 30 850 30 850 100

50 40 130 40 130 5090 100 20

100 30 60 30 60 100

A

800 2500 100 7000 20 47500

90 2500 100 1500 20 10000

350000

175000

2

90 100 20 800

130 50 40 900

60 100 30 850

A

B

C

74

90 800 20

130 900 40

60 850 30

90 100 20

130 50 40

60 100 30

900 40 130 40 130 90090 800 20

850 30 60 30 60 850

175000

90 7000 800 1500 20 56500

175000

700000

175000

4

B

75

90 100 800

130 50 900

60 100 850

90 100 20

130 50 40

60 100 30

50 900 130 900 130 5090 100 800

100 850 60 850 60 100

175000

90 47500 100 56500 800 10000

175000

1925000

175000

11

C

76

1.8 INPUT OUTPUT MODEL

agriculture (A), manufacturing (M),and services (S)

Column A. Production of 1 unit agricultural products

requires the consumption of 0.2 unit of agricultural

products, 0.2 unit of manufactured goods and 0.1 units of

services.

77

78

Example 1.8.1TKK Corporation, a large conglomerate, has three subsidiaries engaged in producing raw rubber, manufacturing tires and manufacturing other rubber based goods. The production of 1 unit of raw rubber requires the consumption of 0.08 unit of rubber, 0.04 units of tires and 0.02 unit of other rubber based goods. To produce 1 unit of tires requires 0.6 unit of raw rubber, 0.02 unit of tires and 0 units of other rubber based goods. To produce 1 unit of other rubber based goods requires 0.3 unit of raw rubber, 0.01 units of tires and 0.06 unit of other rubber based goods. Markets research indicates that the demand for the following year will be RM 200 million for raw rubber, RM 800 million for tires and RM 120 million for other rubber based products. Find the level of production for each subsidiary in order to satisfy this demand.

79

Solution

80

81

82