MATRICESportal.unimap.edu.my/portal/page/portal30/Lecture Notes...Matrix C is not a identity matrix because the matrix is a 3u2 matrix. Example 5 Determine the following matrices are

DKA 104 Preliminary Mathematics

1 Copyright by IMK, UniMAP

Chapter 2 MATRICES

Chapter Outline

2.1 Introduction

2.2 Types of Matrices

2.3 Operations of Matrices

2.4 Determinant of Matrices

2.5 Minor, Cofactor and Adjoint

2.6 The Inverse Matrix

Tutorial & Answers



2.1 INTRODUCTION

If a matrix has m rows and n columns, then the matrix is called nm matrix.

Normally, matrix can be written as nmijaA ][ where ija is denotes the elements i-th

row and j-th column. If ija for ji , then the elements is called the leading diagonal

of matrix A. More generally, matrix A can be written as

nmij

mnmjmmm

inijiii

nj

nj

nj

a

aaaaa

aaaaa

aaaaa

aaaaa

aaaaa

A

][

..

.......

..

.......

..

..

..

321

321

33333231

22232221

11131211

Example 1

Let

364

895

128

A .

Determine

a) order of matrix A

b) elements of the leading diagonal of matrix A

c) elements 23a , 32a and 33a of matrix A

Solutions

a) order of matrix A is 33 .

Definition 2.1

A matrix is an array of real numbers in m rows and n columns

Definition 2.2

If A is an matrix, then A is called a square matrix when the number of

row (i) is equal to the number of column (j).



b) 8, 9 and 3 are elements of the leading diagonal.

c) elements 823 a , 632 a and 333 a .

2.2 TYPES OF MATRICES

Example 2

Determine the following matrices are diagonal matrices or not.

200

580

001

A ,

300

090

008

B ,

300

000

008

C

Solutions

Matrix A is not a diagonal matrix because 023 a .

Matrix B is a diagonal matrix because 0ija and 0iia for ji .

Matrix C is not a diagonal matrix because 022 a .

Example 3

Determine the following matrices are scalar matrix or not.

20

02A ,

400

040

004

B ,

400

020

004

C

Definition 2.3

An matrix is called a diagonal matrix if and for .

Definition 2.4

An matrix is called a scalar matrix if it is a diagonal matrix in which the

diagonal elements are equal, that is and for where k is a

scalar.



Solutions

Matrix A is a scalar matrix where 2iia .

Matrix B is a scalar matrix where 4iia .

Matrix C is not a scalar matrix because the diagonal elements are not equal.

Example 4

Determine the following matrices are identity matrix or not.

1000

0100

0010

0001

A ,

010

001B ,

00

10

01

C

Solutions

Matrix A is a identity matrix because of 1iia .

Matrix B is not a identity matrix because the matrix is a 32 matrix.

Matrix C is not a identity matrix because the matrix is a 23 matrix.

Example 5

Determine the following matrices are zero matrices or not.

000

000

000

A ,

00

00

00

B ,

000

000C

Definition 2.6

An matrix is called a zero matrix or null matrix if all the elements of the

matrix are zero, that is and is written as .

Definition 2.5

An matrix is called identity matrix if the diagonal elements are equal to

1 and the rest of elements are zero, that is and for .

Identity matrix is denoted by .



Solution

Matrices A, B and C are zero matrices because all the elements of the matrices are

zero.

Example 6

Determine the negative matrices of A and B

a)

603

940

1175

A b)

10

34

02

B

Solutions

a)

603

940

1175

A , hence

603

940

1175

A

b)

10

34

02

B , hence

10

34

02

B

Example 7

Determine the following matrices are upper triangular matrix or lower triangular

matrix.

a)

600

940

1175

A b)

1366

07123

0054

0003

B

Definition 2.7

A negative matrix of is denoted by -A where .

Definition 2.8

An matrix is called upper triangular matrix if for .

It is called lower triangular matrix if for .



Solutions

a)

600

940

1175

A is an upper triangular matrix.

b)

1366

07123

0054

0003

B is a lower triangular matrix.

Example 8

Determine the transpose of the following matrices.

a)

603

940

1175

A b)

10

34

02

B

Solutions

a)

6911

047

305TA b)

130

042TB

Definition 2.9

If is an matrix, then the tranpose of A, is the

matrix defined by .



Example 9

Let

763

652

324

A , find TA . Show that the matrix of A is a symmetric matrix?

Solution

AAT

763

652

324

. Hence, A is a symmetric matrix.

Example 10

Let

032

301

210

A , find TA . Shows that the matrix of A is a skew symmetric

matrix?

Solution

AAT

032

301

210

. Hence, A is a skew symmetric matrix.

Definition 2.10

An matrix is called a symmetric matrix if it satisfies , where the

elements obey the rule .

Definition 2.11

An matrix is called a skew symmetric matrix if it satisfies and

the leading diagonal must equal to zero or satisfy this rule for .



Example 11

For the following matrices, determine either the matrices to be in row echelon form or

not. If the matrices aren’t in row echelon form, give a reason.

a)

100

001

001

A b)

000

410

231

B c)

0000

0000

1000

7100

6410

C

Solutions

a) Matrix A isn’t in row echelon form because the number 1 in second

rows appears in the same column.

b) Matrix B is a row echelon form.

c) Matrix C is a row echelon form.

Definition 2.12

An matrix A is said to be in row echelon form (REF) if it satisfies the

following properties :

i) All zero rows, if there any, appear at the bottom of the matrix.

ii) The first nonzero entry from the left of a nonzero is a number 1.

This entry is called a leading ‘1’ of its row.

iii) For each non zero row, the number 1 appears to the right of the

leading 1 of the previous row.

iv) All entries below the leading ‘1’ are zero.

Definition 3.13

An matrix A is said to be in reduced row echelon form (RREF) if it

satisfies the following properties :

i) All zero rows, if there any, appear at the bottom of the matrix.

ii) The first nonzero entry from the left of a nonzero row is a number

1. This entry is called a leading ‘1’ of its row.

iii) For each non zero row, the number 1 appears to the right of the

leading 1 of the previous row.

iv) If a column contains a leading 1, then all other entries in the

column are zero



Example 12

For the following matrices, determine either the matrices to be in reduced row echelon

form or not. If the matrices aren’t in reduced row echelon form, give a reason.

a)

100

010

001

A b)

000

010

001

B c)

100

010

001

B

Solutions

a) Matrix A is in reduced row echelon form.

b) Matrix B is in reduced row echelon form.

c) Matrix C isn’t in reduced row echelon form because 133 a not number 1.

Example 13

Given

3

04

yP and

35

zxQ . If P = Q, find the value of x, y and z.

Solution

Given

353

04 zx

y and the solution is x = 4, y = 5 and z = 0.

2.3 OPERATIONS OF MATRICES

Definition 2.14

Two matrices and are said to be equal and denoted by

A = B if and only if these matrices have the same dimensions and equal

corresponding elements.

Definition 3.15

If and are both matrices, then A + B is an

matrix defined by .



Example 14

Given

364

895

128

A ,

961

365

127

B and

1376

682C .

Find

a) A + B

b) A + C

c) B + C

Solutions

a)

12125

11150

2015

BA

b) A + C (is not possible)

c) B + C (is not possible)

Note: It should be noted that the addition of the matrices A and B is defined only

when A and B have the same number of rows and the same number of columns

Example 15

Given

364

895

128

A ,

961

365

127

B and

1376

682C .

Find

a) A – B

b) B – A

c) B – C

Definition 2.16

If and are both matrices, then the A - B is an

matrix defined by .



Solutions

a)

603

5310

041

BA

b)

603

5310

041

AB

c) B – C (is not possible)

Note: It should be noted that the subtraction of the matrices A and B is defined only

when A and B have the same number of rows and the same number of columns and

ABBA .

Properties of Matrices Addition and Subtraction

If A, B and C are nm matrices, then

a) A + B = B + A

b) A + (B + C) = (A + B) + C

c) A + 0 = A

d) A + (–A) = 0

e) (A + B)T = AT + BT

Example 16

Given

12125

11150

2015

A .

Find

a) 3A b) -2A

Definition 2.17

If is an matrix and k is a scalar, then the scalar multiplication is

denoted by kA where



Solutions

a)

363615

33450

6045

12125

11150

2015

33A

b)

242410

22300

4030

12125

11150

2015

22A

Properties of Scalar Multiplication

If A and B are both nm matrices, k and p are scalar, then

a) kBkABAk )(

b) pAkAApk )(

c) )()( AkppAk

d) (kA)T = kA

T

Example 17

Given

0353

9721A and

742

196

106

722

B .

Find

a) AB b) BA

Definition 2.18

If A is an matrix and B is an matrix, then the product

is the matrix.



Solutions

a) AB =

0353

9721

742

196

106

722

=

233318

799774

b) BA (is not possible)

Properties of Matrix Multiplication

If A, B and C are matrices, I identity matrix and 0 zero matrix, then

a) A(B + C) = AB + AC

b) (B + C)A = BA + CA

c) A(BC) = (AB)C

d) IA = AI = A

e) 0A = A0 = 0

f) (AB)T = BTAT

2.4 DETERMINANT OF MATRICES

Example 18

Find the determinant of matrix

89

52A .

Solution

|A| = 29451689

52

Definition 2.19

If is a matrix, then the determinant of A denoted by

det[A] = |A| and is given by |A| = .



Method of Calculation

3231333231

2221232221

1211131211

21321

aaaaa

aaaaa

aaaaa

LLLLL

- - - + + +

Example 19

Given

334

253

112

P , evaluate |P|.

Solution

34334

53253

12112

- - - + + +

So that, |P| = 2(5)(3) + 1(2)(4) + 1(3)(3) – 1(5)(4) – 2(2)(3) – 1(3)(3)

= 30 + 8 + 9 – 20 – 12 – 9

= 6

Note: Methods used to evaluate the determinant above is limited to only 22 and

33 matrices. Matrices with higher order can be solved by using minor and cofactor

methods.

Definition 2.20

If is a matrix, then the determinant of A is given

by

|A|= .



2.5 MINOR, COFACTOR AND ADJOINT

Example 20

Let

334

253

112

A . Evaluate the following minors

a) 11M b) 12M c) 13M

Solutions

a) 961533

2511 M

b) 18934

2312 M

c) 1120934

5313 M

Example 21

Let

334

253

112

A . Evaluate the following cofactors

a) 11C b) 23C c) 13C

Definition 3.21

Let be an matrix. Let be the submatrix of

A is obtained by deleting the i-th row and j-th column of A. The determinant

is called the minor of A.

Definition 3.22

Let be an matrix. The cofactor of is defined as

.



Solutions

a) 9]615[)1(33

25)1( 21111 C

b) 2]46[)1(34

12)1( 53223

C

c) 11]209[)1(34

53)1( 43113 C

Example 22

Find the determinant of

5224

1119

3026

1423

A by expanding along the

a) first row

b) second row

Definition 3.23

If is cofactor of matrix A, then the determinant of matrix A can be obtained

by

i) Expanding along the i-th row

OR

ii) Expanding along the j-th column



Solutions

a) Expand by the first row,

|A| = 1414131312121111 CaCaCaCa

= 3(–1)1+1

522

111

302

+ (-2)(–1)1+2

524

119

306

+ 4(–1)1+3

524

119

326

+ (1)(–1)1+4

224

119

026

= 3(–6) + 2(–48) +4(–14)+ (–1)(20)

= –190

b) Expand by the second row,

|A| = 2424232322222121 CaCaCaCa

= 6(–1)2+1

522

111

142

+ (2)(–1)2+2

524

119

143

+ 0+ 3(–1)2+4

224

119

423

= 6(38) + 2(–209) +0+3(0)

= –190



Properties of Determinant

a) If A is a matrix, then |A| = |AT|

b) If two rows (columns) of A are equal, then |A| = 0.

c) If a row (column) of A consist entirely of zeros elements, then |A| = 0.

d) If B is obtained from multiplying a row (column) of A by a scalar k, then |B| =

k |A|.

e) To any row (column) of A, we can add or subtract any multiple of any other

row (column) without changing |A|.

f) If B is obtained from A by interchanging two rows (columns), then |B| = –|A|.

g) If A and B are matrices of the same size, then det (AB) = det(A).det(B).

h) Let A be nn matrix and k be a scalar, then )det(kA = ).det(Ak n

Example 23

Let

570

684

312

A , compute adj [A].

Solutions

11C = (–1)1+1

M11 = 57

68 = –2, 12C = (–1)

1+2 M12 = –1

50

64 = –20

13C = (–1)1+3

M13 = 70

84 = 28, 21C = (–1)

2+1 M 21 = –1

57

31 = 16

22C = (–1)2+2

M 22 = 50

32 = 10, 23C = (–1)

2+3M 23 = –1

70

12 = –14

31C = (–1)3+1

M 31 = 68

31 = –18, 32C = (–1)

3+2 M 32 = –1

64

32 = 0

Definition 2.24

Let A is an matrix, then the adjoint of A is defined as adj [A] = [Cij]T.



33C = (–1)3+3

M 33 = 84

12 = 12

We have Cij =

12018

141016

28202

Then, adj [A] = [Cij]T =

121428

01020

18162

2.6 THE INVERSE MATRIX

If AB = I, then A is called inverse of matrix B or B is called inverse of matrix A and

denoted by A = B-1

or B = A-1

.

Example 24

Determine matrix

35

47B is the inverse of matrix

75

43A .

Solution

Note that

75

43AB

10

01

35

47

35

47BA

10

01

75

43

Hence AB = BA = I, so B is the inverse of matrix A (B = A-1

)

Definition 2.25

An matrix A is said to be invertible if there exist an matrix B such

that AB = BA = I where I is identity matrix.



Example 25

Let

46

35A , find A-1.

Solution

A-1

=

2

53

2

32

56

34

2

1.

Example 26

Let

570

684

312

A , find A-1.

Solution

From the example 2.23, adj [A] =

121428

01020

18162

and |A| = 60.

Then,

121428

01020

18162

60

11A

Theorem 1

If and , then A-1

= .

Theorem 2

If A is matrix and , then A-1

= adj [A].



Characteristic of Elementary Row Operations (ERO)

a) Interchange the i-th row and j-th row of a matrix, written as ji bb .

b) Multiply the i-th row of a matrix by a nonzero scalar k, written as kbi.

c) Add or subtract a constant multiple of i-th row to the j-th row, written as

ji bkb or ji bkb

Example 27

By performing the elementary row operations (ERO), find the inverse of matrix

322

253

141

A .

Solution

23322 6:

102

07

1

7

3001

5607

510

141

7

1:

102

013

001

560

570

141

rrRrR

5

7

5

6

5

4

07

1

7

3

07

4

7

5

1007

510

7

1301

5

7:

17

6

7

4

07

1

7

3

07

4

7

5

7

500

7

510

7

1301

4:

17

6

7

4

07

1

7

3001

7

500

7

510

141

33211 rRrrR

5

7

5

6

5

41115

13

5

14

5

11

100

010

001

7

13:

5

7

5

6

5

4111

07

4

7

5

100

0107

1301

7

5: 311322 rrRrrR

Theorem 3

If augmented matrix [ A | I ] may be reduced to [ I | B] by using elementary

row operation (ERO), then B is called inverse of A.

133122 2:

100

013

001

322

570

141

3:

100

010

001

322

253

141

rrRrrR



Then,

57

56

54

513

514

511

1 111A.

TUTORIAL 2

1. Determine the unknowns of the following matrices:

a)

531

423

32

2

wv

u

zyx

b)

fe

dc

ba

32

4

35

68

23

2

c)

5

4

3

2

rqp

qp

p

d) 71362232 zxyxx

e)

z

yx

92113

53

94

311

f)

z

yyx

1032

533

94

52



g)

1211

49

6030

330

142

632

5

347

9101

30155

p

n

m

h)

05

131

0112

075

1331

0119

z

yx

zyx

2. Let

531

423A ,

840

631B and

63

52C .

Determine

a) A + 2B

b) B – A

c) A + 4C

d) 2C – B

e) 3B – 2A

f) 3(A + B)

g) CB

h) BC

i) ATB

j) 2(BAT)

k) (ABT)C

l) A(BTC)

3. If

531

752

321

A and

201

654

123

B , shows that

a) (A + B)T = AT + BT

b) (AB)T = BTAT

c) (5A)T = 5AT



4. For the following matrices, determine whether the matrices are satisfy AB = I or not.

a)

12

25A

52

21B

b)

75

43A

25

47B

c)

432

843

311

A

111

124

458

B

5. Determine the following matrices are symmetric, skew symmetric or non symmetric.

a)

031

305

150

A

b)

6522

5353

2524

2341

B

c)

2521

5554

2524

1341

C

d)

0540

5363

4631

2310

D



6. Which of the following matrices is in REF, RREF or neither.

a)

400

230

321

b)

100

210

321

c)

432

031

001

d)

000

210

301

e)

100

010

001

f)

000

010

021

g)

010

301

421

h)

100

000

201

i)

020

010

001

j)

000

110

351

k)

000

100

431

l)

0000

5100

2310

7641

m)

0000

0000

2100

5031

n)

0000

1000

0100

9071

o)

0000

1000

0410

0301

p)

5100

2610

7341

q)

2010

0100

0301

r)

000000

610000

500310

s)

11000

20100

30031



7. Suppose that 3)det(

ihg

fed

cba

A . Find the following determinant and explain

how you know that value of each determinant is. (Note : You do not have to do any

calculations).

a)

ifc

heb

gda

b)

fed

cba

ihg

c)

fed

ihg

cba 222

d)

ihg

fed

cba

222

222

222

e)

ihg

fed

cba

ihg

fed

cba

det

f)

cibhag

cfbead

cba

222

555

8. Find the determinant of the following matrices:

a)

343239

282619

403829

A b)

70939

53730

23313

B c)

322

211

210

C

9. By imposing the suitable method, find the inverse of the following matrices. If an

inverse does not exist, state this:

a)

706

532

124

A b)

105

84T c)

165

402

321

B



d)

yxy

xxP

2

e)

413

312

111

C

10. Given

706

532

124

P , find

a) Minor of 11a , 12a and 13a .

b) Cofactor of 11a , 12a and 13a .

c) Determinant of matrix P.

d) Adjoin of matrix P.

e) Inverse of matrix P.

ANSWERS

1. a)

592

1413

wvu

zyx b)

24

232

35

fe

cd

ba

c)

18

,10

,3

r

q

p

d)

5

,2

,3

z

y

x

e)

12

,8

,14

z

y

x

f)

0

,17

,24

z

y

x

g)

7

,10

,5

p

n

m

h)

7

,10

,13

z

y

x



2. a)

2151

1641 b)

371

254

c) is not possible d) is not possible

e)

14182

10139 f)

3933

3036

g)

66153

52142 h) is not possible

i)

6484

12182

10133

j)

5680

8030

k)

324

16590 l)

324

16590

3. a) TTT BABA )( ,

7134

300

264

732

1306

404T

b) TTT ABAB )( ,

274217

13218

14218

271314

422121

1788T

c) TT AA 5)5( ,

253515

152510

5105

25155

352510

15105T

4. a) IAB b) IAB c) IAB



5. a) skew symmetric b) symmetric

c) non symmetric d) non symmetric

6. a) neither b) REF c) neither

d) RREF e) RREF f) REF

g) neither h) neither i) neither

j) REF k) REF l) REF

m) RREF n) REF o) RREF

p) REF q) neither r) RREF

s) RREF

7. a) 3 b) 3 c) -6

d) 24 e) 9 f) 3

8. a) 360 b) 1 c) 15

9. a)

11

4

11

6

11

9112

22

13

11

7

22

21

1A

b) no inverse because the determinant is zero

c)

14

1

14

1

14

328

1

7

2

28

117

1

14

5

7

3

1B

d) no inverse because the determinant is zero

e)

9

1

9

2

9

19

1

9

7

9

179

2

9

5

9

7

1C



10. a) 2111 M , 4412 M , 1813 M

b) 2111 C , 4412 C , 1813 C

c) 22P

d)

81218

222244

131421T

ijCPAdj

e)

11

46

11

9112

22

13

11

7

22

21

1P

Documents

MATRICESportal.unimap.edu.my/portal/page/portal30/Lecture Notes...Matrix C is not a identity matrix because the matrix is a 3u2 matrix. Example 5 Determine the following matrices are