BA201 Engineering Mathematic UNIT7 - Matrices Operation

B3001/UNIT7/1

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Prepared by : Nur HIdayah Othman Page 1

Unit

7

MATRICES OPERATION

To know the different types of matrices and

understand how to apply it on simple algebra

problem solving.

Upon completing this module, you should be

able to:

1. Identify and determine the

determinant from the square matrix.

2. Identify and determine the minor

matrix from the square matrix.

3. Identify and determine the cofactor


4. Identify and determine the inverse


General Objectives

Specific Objectives

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7.0 INTRODUCTION

There are a few methods to solve simultaneous equations using the matrices such as Cramer’s

Rule and Inverse Matrices. Before we go further to look at the Cramer’s Rule and Inverse Matrices, we

should know and understand the operations of the matrices like determinant of the matrix, minor

matrices, co-factor, adjoint and inverse of square matrices.

7.1 DETERMINANT OF THE MATRIX

Determinant is a unique number that can be determined from the square matrix. It is used to

represent the real-value of the matrix which can be used to solve simple algebra problems later

on.

The symbol for the determinant of matrix A is det(A) or A.

7.1.1 Determinant of the matrix 2 x 2

For a matrix of size 2 x 2, the method to find the determinant is:

Let’s say, A =

dc

ba

Then, det(A) = A = dc

ba = (ad – bc)

Example 7.1:

If A =

87

65 determine det(A).

INPUT

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Solution:

A = (58 - 67)

= 40 – 42

= -2

Example 7.2:

If A =

95

24 determine det(A).

Solution:

A = (4·9 - 2·5)

= 32

= 12

7.1.2 Determinant of The Matrix 3 x 3

For a matrix of size 3 x 3, the method to find the determinant is:

If A =

333231

232221

131211

aaa

aaa

aaa

Then, A= 3231

2221

31

3313

2312

21

3332

2322

11aa

aaa

aa

aaa

aa

aaa

A= 312232213113233312213223332211 aaaaaaaaaaaaaaa

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Example 7.3 :

Determine of the determinant of matrix

212

034

231

Solution:

Determinant = 12

342

22

043

21

031

= 1(3·2 – 0·1) – 3(4·2 – 0·2) + 2(4·1 – 3·2)

= 6 – 24 – 4

= –22

Example 7.4 :

If A =

864

297

531

, determine A.

Solution:

A = 64

975

84

273

86

291

= 1(9·8 – 2·6) – 3(7·8 – 2·4) + 5(7·6 – 9·4)

= 60 – 144 + 30

= –54

B3001/UNIT7/5

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7a.1 Determine the determinant for the following matrices.

a.

124

136

b.

127

401

351

c.

35

83

7a.2 If A =

724

432

612

B =

36

24 dan C =

052

640

241

Determine:

i) A

ii) B

iii) C

ACTIVITY 7a

B3001/UNIT7/6

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FEEDBACK 7a

7a.1 a. 20

b. 1

c. -49

7a.2 i). 4

ii). 24

iii). -46

B3001/UNIT7/7

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7.2 MATRIX MINOR

The minor of a matrix is a new matrix where all the elements are determinants. Each

determinant is calculated by removing a row and a column from the original matrix. For

example, in order to determine the element at position ij, you will have to remove row i and

column j from the matrix. Next, you calculate the determinant of what is left.

If, A =

333231

232221

131211

aaa

aaa

aaa

then, Minor A =

333231

232221

131211

MMM

MMM

MMM

Where,

3332

2322

11aa

aaM by removing row 1 and column 1 from A

3331

2321

12aa


3231

1211

23aa


INPUT

B3001/UNIT7/8

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Example 7.5:

If A =

864

297

531

, determine minor A.

Solution:

The elements are:

86

2911 M = 9·8 – 2·6 = 60

84

2712 M = 7·8 – 2·4 = 48

64

9713 M = 7·6 – 9·4 = 6

86

5321 M = 3·8 – 5·6 = -6

84

5122 M = 1·8 – 5·4 = -12

64

3123 M = 1·6 – 3·4 = -6

29

5331 M = 3·2 – 5·9 = -30

27

5132 M = 1·2 – 5·7 = -33

97

3133 M = 1·9 – 3·7 = -12

Therefore, Minor A =

123330

6126

64860

B3001/UNIT7/9

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Example 7.6:

If P =

212

034

231

, determine minor P.

Solution:

The elements are:

21

0311 M = 3·2 – 0·1 = 6

22

0412 M = 4·2 – 0·2 = 6

12

3413 M = 4·1 – 3·2 = -2

21

2321 M = 3·2 – 2·1 = 4

22

2122 M = 1·2 – 2·2 = -2

12

3123 M = 1·1 – 3·2 = -5

03

2331 M = 3·0 – 2·3 = -6

04

2132 M = 1·0 – 2·4 = -8

34

3133 M = 1·3 – 3·4 = -9

Therefore, Minor P =

986

524

266

B3001/UNIT7/10

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7b.1 Determine the Minor of the following matrices:

i) A =

724

432

612

ii) B =

127

401

351

iii) C =

052

640

241

ACTIVITY 7b

B3001/UNIT7/11

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FEEDBACK 6b

7b.1 i)

82014

0105

163013

ii)

5120

372011

2278

iii)

4616

13410

81230

B3001/UNIT7/12

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7.3 CO-FACTOR OF A MATRIX

Once you have found the Minor of a matrix, you can easily determine the Cofactor of the matrix.

All the hard work is already done when you determine the Minor of a matrix. All you need to do

now is to multiply each element of the Minor of the matrix with a factor ji1 and the Cofactor

is done.

Let’s look at the following example:

If A =

333231

232221

131211

aaa

aaa

aaa

and Minor A =

333231

232221

131211

MMM

MMM

MMM

Therefore, Co-factor of a matrix A =

333231

232221

131211

KKK

KKK

KKK

Where, ij

ji

ij MK )1(

Then,

Co-factor of a matrix A =

33

6

32

5

31

4

23

5

22

4

21

3

13

4

12

3

11

2

111

111

111

MMM

MMM

MMM

INPUT

B3001/UNIT7/13

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Example 7.7:

If A =

864

297

531

, determine the co-factor of a matrix A

Solution:

First, find the minor of a matrix A,

123330

6126

64860

Next, multiply each element by its factor ji1

Therefore, the co-factor of a matrix A,

A =

121331301

6112161

61481601

654

543

432

A =

123330

6126

64860

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Example 7.8:

If P =

212

034

231

, determine the co-factor of a matrix P.

Solution:

First, find the minor of a matrix P,

986

524

266

Next, multiply each element by its factor ji1

Therefore, the co-factor of a matrix,

P =

918161

512141

216161

654

543

432

P =

986

524

266

B3001/UNIT7/15

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7c.1 Find the cofactor for the following matrices:

i) A =

724

432

612

ii) B =

127

401

351

iii) C =

052

640

241

AKTIVITY 6c

B3001/UNIT7/16

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FEEDBACK 7c

7c.1 i)

42014

0105

163013

ii)

5120

37201

2278

iii)

4632

13410

81230

B3001/UNIT7/17

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7.4 ADJOINT MATRIX

For a square matrix A with n x n, you can find the adjoint of a matrix when transposing the

cofactors of a matrix A.

In this case, for matrix A,

Adjoint of a matrix A, written as Adj(A) = KT where K is the co-factor for A

Then, if A =

333231

232221

131211

aaa

aaa

aaa

and Minor A =

333231

232221

131211

MMM

MMM

MMM

And co-factor matrix A =

333231

232221

131211

KKK

KKK

KKK

Then adjoint matrix A, Adj(A) =

332313

322212

132111

KKK

KKK

KKK

Or,

TKAAdj )(

INPUT

B3001/UNIT7/18

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Example 7.9:

If A =

864

297

531

, determine the ad joint of the matrix A.

Solution:

Inputs from example 4.14, you will find minor for A =

123339

6126

64860

And cofactor of A =

123339

6126

64860

Then, adjoint of matrix A, Adj(A) =

1266

331248

39660

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Example 7.10:

If P =

212

034

231

, determine the ad joint for matrix P.

Solution:

Inputs from example 4.15, the minor of P =

976

524

286

and co-factor of P =

976

524

286

with the adjoint for matrix P; Adj(P) =

952

728

646

B3001/UNIT7/20

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7D.1 Determine the adjoint for the following matrices:

i) A =

724

432

612

ii) B =

127

401

351

iii) C =

052

640

241

ACTIVITY 7d

B3001/UNIT7/21

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FEEDBACK 7d

7d.1 i)

4016

201030

14513

ii)

5372

12027

2018

iii)

4138

6412

321030

B3001/UNIT7/22

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7.5 INVERSE OF A SQUARE MATRIX

The inverse of a square matrix is its complement because when you multiply a matrix with its

inverse, the product is an Identify matrix.

In other words, if A is a square matrix and A-1

its inverse, then

AA-1

= I , where I is an Identity matrix.

In order to find A-1

, you can make use of the Adj(A) and A. The formula to find A-1

is

A-1

= A

1Adj(A)

Example 7.11:

If A =

864

297

531

, find A-1

.

Solution:

The determinant of A,

A = 64

975

84

273

86

291

= 60 – 144 + 30

= –54

INPUT

B3001/UNIT7/23

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The adjoint of A,

Adj(A) =

1266

331248

39660

Therefore, A-1

= A

1Adj(A)

= 54

1

1266

331248

39660

A-1

=

54

12

54

6

54

654

33

54

12

54

4854

39

54

6

54

60

Example 7.12:

If P =

212

034

231

, find P-1

.

Solution:

The determinant of P,

P = 12

342

22

043

21

031

= 1(3·2 – 0·1) – 3(4·2 – 0·2) + 2(4·1 – 3·2)

= 6 – 24 – 4

= –22

B3001/UNIT7/24

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The adjoint for matrix P,

Adj(P) =

952

728

646

Therefore, P-1

= P

1Adj(P)

= 22

1

952

728

646

P-1

=

22

9

22

5

22

222

7

22

2

22

822

6

22

4

22

6

B3001/UNIT7/25

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7e.1 Find the inverse of the following matrices:

i) A =

724

432

612

ii) B =

127

401

351

iii) C =

052

640

241

ACTIVITY 7e

B3001/UNIT7/26

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FEEDBACK 7e

7e.1 i)

40

4

40

0

40

1640

20

40

10

40

3040

14

40

5

40

13

ii)

121

5

121

37

121

2121

1

121

20

121

27121

20

121

1

121

8

iii)

142

4

142

13

142

8142

6

142

4

142

2142

32

142

10

142

30

B3001/UNIT7/27

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SELF ASSESSMENT

7.1 If A =

071

526

341

determine A .

7.2 If A =

071

526

341

determine minor for A.

7.3 If A =

071

526

341

determine cofactor for matrix A.

7.4 If A =

071

526

341

determine adjoint for matrix A.

7.5 If A =

071

526

341

determine the inverse for matrix A.

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SOLUTIONS TO SELF ASSESSMENT

7.1 105

7.2

221314

3321

40535

7.3

10813

5520

15624

7.4

10515

856

132024

7.5

45

10

45

5

45

1545

8

5

5

45

645

13

45

20

45

24

Documents

BA201 Engineering Mathematic UNIT7 - Matrices Operation