Matrix Algebra

Prepared by Deluar Jahan Moloy

Lecturer

Northern University Bangladesh

MatrixA matrix is a rectangular or square array of numbers or values arranged in rows and columns enclosed by a pair of brackets or a pair of double bars or a pair of parentheses. Matrix are generally denoted by capital letters such as A, B, C…..

For examples,

762

3510,

963

852

741

,53

42

WXA

Rows and Columns

The horizontal and vertical lines in a matrix are called rows and columns respectively.

For example

Row

column

53

42

Elements

The numbers or values in the matrix are called its entries or its elements.

For example,

Here 10, 5, 3, 2, 6 and 7 are called elements.

762

3510

Dimension or its order

The number of rows and columns that a matrix has is called its dimension or its order. It is denoted by .Where m represents no. of rows and n represents no. of columns.

By convention, rows are listed first and columns are second.

nm

Examples

We would say that the dimension (or order) of the 3rd matrix is 2 x 3, meaning that it has 2 rows and 3 columns.

3233

22762

3510,

963

852

741

,53

42

Row matrix

A matrix with only one row (a 1 × n matrix) is called a row matrix.

For example,

A = is a 1 × 4 matrix. 2,0,20,1

Column matrix

A matrix with one column (an m × 1 matrix) is called a column matrix.

For example,

A = is a 4 × 1 matrix.

8

6

6

5

Square matrix

A matrix is said to be square matrix if no. of rows and no. of columns are equal.

For example,

is a 2× 2 square matrix.

is a 3 × 3 square matrix.

2253

42

33963

852

741

Unit or Identity matrix

A square matrix whose diagonal elements are unity (or one) and the remaining elements are zero is called Identity matrix.

For example

33100

010

001

Null or Zero Matrix

A matrix each of whose elements are zero is called zero or null matrix.

For example,

are null or zero matrices.32

33

000

000

000

000

000

and

Transpose Matrix

The transpose of one matrix is another matrix

that is obtained by using rows of the first matrix

as columns of second matrix. For example,

Then transpose of a is denoted by

db

caA

dc

baAc

Equal matrices

Two matrices are said to be equal if and only if

1. They are of the same order, i.e. they

have the same number of rows and

columns and

2. Each element of one is equal to the

corresponding elements of the other.

Equal matrix

For example,

and are equal matrices. 22

15

82

22

15

82

Exercise 1 Given

A= B=

If A=B, find the value of a, b, c, d and e.

439

546

291

e

a

dc

39

26

1

Exercise 2 Find the value of x and y if

74

23

4

2

yx

yx

Addition of matrices

To add two matrices A and B: # of rows in A = # of rows in B # of columns in A = # of columns in B

The sum is obtained by adding each element of A with corresponding element of B.

Subtraction of matrices

To subtract two matrices A and B: # of rows in A = # of rows in B # of columns in A = # of columns in B

The difference is obtained by subtracting each element of B from the corresponding element of A.

Addition and subtraction of matrices

If

Then

A + B = possible or not

C + B = possible or not

A + C = possible or not

A - B = possible or not

C - B = possible or not

A - C = possible or not

36

62

853

1066

901

,53

61CandBA

Addition and subtraction of matrices

Form previous,

A + C = + =

=

53

61

36

62

99

01

3363

6621

Addition and subtraction of matrices

Form previous,

A - C = - =

=

53

61

36

62

03

6621

3363

)6(6)2(1

Addition

1

4

2

3

5

8

6

7+ =

A B+ =

Exercise1 If

A= B=

Find i) A+B

ii) A-B

439

546

291

622

020

248

Scalar Multiplication of a matrix

A real number is referred to as a scalar when it occurs in operations involving matrices.

If A is an m × n matrix and k is a scalar, then we let kA denote the matrix obtained by multiplying every element of A by k.

Examples

If A=

Find 2A 2A=2.A=2. =

527

074

527

074

10414

0148

5.2)2.(27.2

0.27.24.2

Exercise1

If A= B=

Find the value of 5A+3B

412

320

541

367

Multiplication of Matrices

If A and B are two matrices such that the number of

columns (n) in A is equal to the number of rows

(m) in B, then the product of A and B denoted by

AB. (The number of columns (n) in first matrix

must equal the number of rows (m) in 2nd matrix in

order to carry out the matrix multiplication.)

Multiplication of Matrices

To multiply two matrices A and B # of columns in A (1st matrix) = # of rows in

B (2nd matrix)

If and . Then matrix multiplication is possible. The product is denoted by AB=A.B.

nmA pnB

Example1

If B= C=

BC=B.C=

2213

20

2241

67

41

67

13

20

67

2018

14213401

16273607

Example2

If A= , find AB

BA. Is AB = BA?

23

11

31

52Band

Example 3

If A=

Find AB and BA if they are possible.

5013

2102

4321

101

123

012

Band

Example4

101

123

012

A

Given that

Find

AA

AA

5.2

3.12

2