BA201 Engineering Mathematic UNIT6 - Matrices Introduction

B3001/UNIT6/1

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Prepared by : Nur Hidayah Othman

Unit

6

MATRICES (Introduction and types of matrices; problem

solving on simultaneous equation using

matrices method)

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:

Define matrix and identify the notation used

in matrices.

Solve matrix problems that involve simple

matrix algebra operations.

Define and determine the size of a matrix, a

square matrix, a symmetrical matrix,

diagonal matrix, identity matrix, and

transpose of a matrix.

General Objectives

Specific Objectives

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6.0 INTRODUCTION

Matrices are sets of numbers that are arranged in rectangular forms. It is a

rectangular array of numbers. These numbers are arranged inside a round or

square bracket. Look at the examples shown below.

8324

4

5

2

325

793

32

41

69

16

It is important to study the fundamentals of matrices first and get a good

introduction on how to apply simple algebra operations on matrices. This can

help in solving engineering problems. For example, you can use matrices to solve

systems of linear simultaneous equations.

6.1 FUNDAMENTALS OF MATRIX

Each number inside a matrix is called an element of the matrix. These numbers

are arranged in rows and columns.

Rows are the horizontally arranged elements of the matrix

For example, the shaded region in the matrix below is the second row of the

matrix.

252

620

430

441

INPUT

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Columns are the vertically arranged elements of the matrix.

For example, the shaded region in the matrix below is the second column of the

matrix.

252

620

430

441

Note: It is common practice to use bold capital letters or underlined capital letter

like A or A to represent a matrix, and small letters to represent the elements.

6.2 SIZE OF A MATRIX

The size of a matrix is the number of rows and columns that it has. If a matrix has

3 rows and 4 columns, then its size is 3 x 4.

Let’s look at the following matrix.

A =

6935

1038

7441

How many rows and columns do you see?

Do you agree that the size of matrix A is 3 x 4?

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Example 6.1:

State the size of the following matrix.

047

933

663

194

Solution:

There are 4 rows and 3 columns. Therefore, the size of this matrix is 4 x 3.

For a matrix A of size 3 x 4, you can use the notation A34 to represent the matrix.

In general, any matrix can be represented by the notation matrix Aik with

i = 1, 2, 3, ….

k =1, 2, 3, ……

The first subscript, i, represents the rows and the second subscript, j, represents

the columns.

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ACTIVITY 6a

6.1 State the size of each of the following matrices

a. 32

b.

5

4

3

c.

672

413

201

d.

75

91

23

6.2 Referring to matrix B =

271

480

353

, state the element at

a. b23

b. b21

c. b31

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Feedback for Activity 6a

6.1

a. 1 x 2

b. 3 x 1

c. 3 x 3

d. 3 x 2

6.2

a. b23 = 4

b. b21 = 0

c. b31 = -1

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6.3 ADDITION, SUBTRACTION AND MULTIPLICATION OF MATRIX

6.3.1 MATRIX ADDITION

The first algebra operation we are going to learn is how to add two matrices. Matrix

addition can only be performed on matrices that have the same size. The result of a

matrix addition is a new matrix that is of the same size. All we need to do is to match the

elements that are at the same position in their matrices. For example, element 12a should

be matched to element 12b to produce element 12c .

If A =

mnm

n

n

aa

aaa

aaa

..........

::::

:

:

1

22221

11211

and B =

mnm

n

n

bb

bbb

bbb

..........

::::

:

:

1

22221

11211

so C = A + B

=

mnmnmm

nn

nn

baba

bababa

bababa

........

::::

....

....

11

2222222121

1112121111

INPUT

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Example 6.3 :

Given that,

3517

2331

10413

A dan

1031

0314

0852

B

determine A + B.

Solution:

)1(305)3(11)7(

0)2()3(313)4(1

010)8(45)1()2(3

BA

A + B =

2526

2043

10441

6.3.2 MATRIX SUBTRACTION

Like matrix addition, matrix subtraction can only be performed on matrices that

have the same size. The result of a matrix subtraction is a new matrix that is of

the same size too. Similarly,

If A =

mnm

n

n

aa

aaa

aaa

..........

::::

:

:

1

22221

11211

and B =

mnm

n

n

bb

bbb

bbb

........

::::

....

....

1

22221

11212

Then, C = A B =

mnmnmm

nn

nn

baba

bababa

bababa

........

::::

....

....

11

2222222121

1112121111

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Example 6.4 :

Given that,

3517

2331

10413

A and

1031

0314

0852

B

determine A – B.

Solution:

Then,

)1(305)3(11)7(

0)2()3(313)4(1

010)8(45)1()2(3

BA

4548

2625

101265

BA

6.3.3 MATRIX MULTIPLICATION

Matrix multiplication is a little bit more complicated. In order to be able

to multiply two matrices AB, we have to ensure that the number of columns in

matrix A is the same as the number of rows in matrix B. That means we can

multiply matrix Amn with matrix Bnk because matrix A has n columns and matrix

B has n rows too. The result is a new matrix that has m rows and k columns.

The actual multiplication process involves taking a row i from matrix A

and matching it with a column j from matrix B. The result becomes the element

ij of the new matrix.

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The operation is as follows:

If A =

mnm

n

n

aa

aaa

aaa

..........

::::

:

:

1

22221

11211

and B =

jkj

k

k

bb

bbb

bbb

........

::::

....

....

1

22221

11211

matrix m x n matrix j x k

then, C = AB with size m x k

Then, C =

mkm

k

k

cc

ccc

ccc

........

::::

....

....

1

22221

11211

where c11 = a11 x b11 + a12 x b21 + a13 x b31+ ……… + a1n x bj1

c12 = a11 x b12 + a12 x b22 + a13 x b32 + ……… + a1n x bj2

c21 = a21 x b11 + a22 x b21 + a23 x b31 + ……… + a2n x bj1

.

.

.

cmk = am1 x b1k + am2 x b2k + am3 x b3k + ……… + amn x bjk

Remember!

Multiplication can only

happen if n = j

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Example 6.5:

Find the multiplication of

10

25A and

11

73B

Solution:

11

73

10

25AB =

)1(17011)3(0

)1(27512)3(5

=

1010

235215

AB =

11

3313

Example 6.6:

Find the products of matrix A and B, given that,

126

410

872

101

A and

54

13

02

B

Solution:

The size of matrix A is 4 x 3 and the size of matrix B is 3 x 2. Therefore we can

multiply them. The new matrix will be of size 4 x 2.

51)1(20641)3(226

5)4()1(1004)4()3(120

58)1()7(0248)3()7(22

51)1(00141)3(021

AB

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5204612

20101630

407032214

500402

310

2119

4757

56

AB

Note: We cannot find the product of BA because the number of columns in matrix B is

not the same as the number of rows in matrix B.

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ACTIVITY 6b

6b1 Based on the following matrices,

A =

59

73 B =

53

43

C =

6

2 D =

75

24

determine:

a. A + B

b. A – C

c. D + (B – A)

d. B + C

6b2 Given that A =

14

12 and B =

32

10,

find:

a. AB

b. BA

6b3 If M =

271

480

353

and N =

5

1

2

,

Find the product of MN.

B3001/UNIT6/14

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Feedback for Activity 6b

6b1

a.

06

116

b. no solution.

c.

37

54

d. no solution.

6b2

a.

12

12

b.

18

14

6b3 MN =

15

28

4

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6.4 TYPES OF MATRICES

6.4.1 SQUARE MATRIX

A square matrix is a matrix where the number of rows is equal to the number of

columns. The following examples are square matrices.

53

43

271

480

353

15491

317210

231254

91623

6.4.2 DIAGONAL MATRIX

If all the elements of a square matrix consist of zeros except the diagonal, then

this matrix is called a diagonal matrix. The following examples are diagonal

matrices.

33

22

11

00

00

00

a

a

a

50

03

200

080

003

6.4.3 IDENTITY MATRIX

If all the elements of a diagonal matrix consist of the value 1, then the matrix is an

identity matrix. The following examples are identity matrices.

I =

100

010

001

I =

10

01 I =

1000

0100

0010

0001

INPUT

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An identity matrix is special because when you multiply a matrix with it or when

you multiply it with a matrix, the matrix does not change. For example:

AI = A IB = B MI = M

6.4.4 TRANSPOSE OF MATRIX

When you interchange the rows of a matrix with its columns, you would have

converted a matrix Amn to another matrix Anm. In other words, a matrix of size m

x n now is size n x m. This new matrix is called the transpose of a matrix. The

symbol for a transpose of a matrix A is AT.

Let’s look at the following example.

If A =

3231

2221

1211

aa

aa

aa

then, AT =

232221

131211

aaa

aaa

Example 5.7:

If A =

106

612

002

then, transpose matrix of A, AT =

160

010

622

Shown that, the first row becomes the first column, the second row becomes the second

column and so on.

Transpose for the transpose of matrix given the original matrix A

(AT)T = A

Some important properties relating to transpose are:

(AB)T = B

TA

T

(ABC…Z)T = Z

T…..B

TA

T

(A + B)T = A

T + B

T

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6.4.5 SYMMETRIC MATRIX

When you interchange the rows of a matrix with its columns, you would have

converted a matrix Amn to another matrix Anm. In other words, a matrix of size m

x n will now be of size n x m. This new matrix is called the transpose of a matrix.

The symbol for a transpose of a matrix A is AT. Let’s look at the following

example.

A = AT (symmetric)

Example 6.8:

Determine whether the following matrices are symmetric or not.

a)

632

321

b)

305

028

681

c)

453

502

321

d)

937

326

761

Solution:

a) not symmetric

b) not symmetric

c) is symmetric

d) is symmetric

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AKTIVITI 6c

6c.1 Observe the following matrices:

A =

50

03 B =

23

42

C =

001

010

100

D =

3

5

5

3

E =

864

642 F =

10

01

Determine of the above matrices are of the following types:

a) Diagonal matrix

b) Square matrix

c) Symmetric matrix

d) Identity matrix

6c.2 If A =

03

34

21

, determine AT.

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Feedback for Activity 6c

6c.1

a) Diagonal matrix: A and F

b) Square matrix: A, B, C and F

c) Symmetric matrix: C and F

d) Identity matrix: F

6c.2 AT =

032

341

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SELF ASSESSMENT

6.1 Given two matrices A and B, where

A =

70

52

31

and B =

21

31

02

,

evaluate:

a. A + B

b. A – B

6.2 If A =

95

37

21

and B =

271

480

353

,

evaluate:

a. BA

b. AI

c. IB

d. B2

(Note: B2 = B x B )

6.3 Given that

A =

19

03

72

and B =

421

039,

Prove that (AB)T = B

T A

T.

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SOLUTIONS TO SELF ASSESSMENT

6.1 a) A + B =

91

81

33

b) A – B =

51

23

31

6.2 a)

3758

6076

623

b) A

c) B

d)

35655

40924

53412

6.3 AB =

42980

0927

28825

(AB)T =

4028

2998

802725

BT A

T =

4028

2998

802725

Therefore, (AB)T = B

T A

T.

BA201 Engineering Mathematic UNIT6 - Matrices Introduction

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