Prepared by:Sim Win NeeKeu Pei San

Choon Siang YongCh’ng Tje Yie

4.7 INVERSE MATRIXPrepared By:

Sim Win Nee

Keu Pei San

Choon Siang YongCh’ng Tje

Yie

Inverse matrixThe inverse of a

square matrix is another matrix such

that when the two are multiplied together, in any order, the product is an identify matrix.

Inverse matrix

When a number , n , multiplies by its reciprocal n¯¹ , the product is 1.

The inverse of a matrix, A, is denoted by A¯¹ and the product of A x A¯¹ is the identify matrix, I .

Example:

Determine whether matrix A is the inverse of matrix B.

a) A=[3 45 7 ] , B=[ 7 -4

-5 3]b) A=[2 5

3 8] ,B=[ 8 53 2]

Solution:

a)AB=[3 45 7 ][ 7 -4

-5 3 ] = [21+(-20) -12+12

35+(-35) -20+21 ] = [1 0

0 1]

BA = [ 7 -4-5 3][3 4

5 7] = [ 21+(-20) 28+(-28)

-15+15 -20+21 ] = [1 0

0 1]AB= I and BA= I

Therefore; A is the inverse matrix of B, A=B ¹.ˉ B is the inverse matrix of A, B=A ¹.ˉ

b) AB= [ 2 53 8 ][ 8 5

3 2] = [ 16+15 10+10

24+24 15+16 ] = [31 20

48 31] ( Not equal to I )

Therefore; A is not the inverse matrix of matrix B, A is not equal to B.

Exercise:

Determine whether the matrix A and B are the inverses of one another.

1. A= [ 3 -2-4 3] , and B= [3 2

4 3] 2. A= [5 3

8 5] , and B= [5 -38 5]

INVERSE MATRIX

4.7 B Determining the Inverse of a 2x2 Matrix

There are two methods to find the inverse of a matrix,a) by using the method of solving simultaneous linear equation

b) By using formula

a. Method of solving simultaneous equations

Given, matrix A = [ ]To find the inverse of matrix A, let A¯¹ = [ ] A x A¯¹ = I

Then; [ ][ ]=[ ]

[ ]=[ ] Equal Matrices

3 13 4

a b c d

3 13 4

a bc d

1 00 1

3a + c 3b + d 3a + 4c 3b + 4d

1 00 1

3a + c = 1 1 3b + d = 0 3

3a + 4c = 0 2 3b + 4d = 1 4

1-2 : -3c = 1 3-4 : -3d = -1

c = - d =

Substitute c = - in equation 1

3a + (- ) = 1

a =

Substitute d = in equation 3

3b + ( ) = 0

b = -

Therefore, Aˉ¹=[ ]Check the answer; AAˉ¹= [ ][ ] = [ ]= I

3 13 4

1 00 1

3

1

3

1

3

1

3

1

9

4

3

1

3

1

9

1

9

4

3

1

9

1

3

1

9

4

3

1

9

1

3

1

Given the matrix B, find the inverse B ¹by using the method of solving ˉsimultaneous linear equations.

B= [ ]Solution: Let B ¹= ˉ [ ]

[ ][ ]=[ ] [ ]=[ ]

4e + 3g = 1 1 4f + 4h = 0 34e + 4g = 0 2 4f + 4h = 1 4

4 34 4

e fg h

4 34 4

e fg h

1 00 1

4e +3g 4f +3h4e +4f 4f +4g

1 00 1

EXAMPLE 1

2-1 : g = -1 4-3 : h = 1So, 4e + 3(-1) = 1 so, 4f + (1)= 0 e = 1 f = -

Therefore, B ¹= ˉ [ ]

4

3

4

31

1 1

B. USING FORMULA

We can obtain the inverse of 2 x 2 matrix by using the following formula.

In general, if A = [ ]The inverse of matrix A is

Aˉ¹ = [ ] [ ]ad-bc is the determinant and written as |A|

a bc d

d -b-c a

bcad 1

bcadb

bcad

d

bcada

bcad

c

Example 2Find the inverse of the , by using the formula

a) G =[ ]Determinant, |G|= ad – bc

= (4x2)-(3x2) = 2

Therefore, G ¹= ˉ [ ] = [ ]

4 32 2

2 -3-2 4

2

1

1

12

3

2

1. Using the method of solving simultaneous equations, find the inverse matrix for each of the matrices given below.

a)B=[ ]

2. Find the inverse matrix for each of the matrices given below using formula.

a) B= [ ]

9 75 4

3 7-1 -3

The End…..