1. 4.7 INVERSE MATRIX Prepared By: Sim Win Nee Keu Pei San Choon Siang Yong Ch’ngTjeYie Prepared by:Sim Win NeeKeu Pei SanChoon Siang YongCh’ngTjeYie

2. Inverse matrix The 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.

3. 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 .

4. Example:Determine whether matrix P is the inverse of matrix Q. a) A=[ ] [ ] 7 -4 -5 3 4 5 7 , B= b) A=[ ] ,B=[ ] 5 3 8 5 3 2

5. Solution: a)AB=[ ][ ] 4 5 7 7 -4 -5 3 = [ ] 21+(-20) -12+12 35+(-35) -20+21 = [ ] 0 0 1

6. BA = [ ][ ] 7 -4 -5 3 3 4 5 7 = [ ] 21+(-20) 28+(-28) -15+15 -20+21 = [ ] 0 0 1 AB= I and BA= I Therefore; A is the inverse matrix of B, A=B-1. B is the inverse matrix of A, B=A-1.

7. b) AB= [ ][ ] 5 3 8 8 5 3 2 = [ ] 16+15 10+10 24+24 15+16 = [ ] 20 48 31 ( Not equal to I ) Therefore; A is not the inverse matrix of matrix B, A is not equal to B.

8. Exercise:Determine whether the matrix A and B are the inverses of one another. 1. A= [ ] , and B= [ ] 3 -2 -4 3 2 3 2. A= [ ] , and B= [ ] 3 8 5 -3 8 5

10. 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 1 3 4 a b c d 0 0 1 1 3 4 a b c d 3a + c 3b + d 3a + 4c 3b + 4d 0 0 1

11. 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 = -⅓ d = ⅓ Substitute c =-⅓ in equation 1 3a + (-⅓) = 1 a = 4⁄9 Substitute d = ⅓ in equation 3 3b + (⅓) = 0 b = - 1/9 Therefore, A⁻1=[ ] Check the answer; AA⁻1 = [ ][ ] = [ ]= I 4/9 -1/9 -1/3 1/3 1 3 4 4/9 -1/9 -1/3 1/3 0 0 1

12. Example 1 Given the matrix B, find the inverse B¯1 by using the method of solving simultaneous linear equations. B= [ ] Solution: Let B¯1= [ ] [][ ]=[ ] [ ]=[ ] 4e + 3g = 1 1 4f + 4h = 0 3 4e + 4g = 0 2 4f + 4h = 1 4 3 4 4 e f g h 3 4 4 0 0 1 e f g h 4e +3g 4f +3h 4e +4f 4f +4g 0 0 1

13. 2-1 : g = -1 4-3 : h = 1 So, 4e + 3(-1) = 1 so, 4f + (1)= 0 e = 1 f = -3⁄4 Therefore, B⁻1 = [ ] -3⁄4 -1 1

14. 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 A⁻1 =1⁄ad – bc[ ] [ ] ad-bc is the determinant and written as |A| a b c d d -b -c d d/ad – bc -b/ ad-bc -c/ ad-bc a/ad-bc

15. Example 2 Find the inverse of the , by using the formula a) G =[] Determinant, |G|= ad – bc = (4x2)-(3x2) = 2 Therefore, G⁻1 =1/2 [ ] = [ ] 3 2 2 -3 -2 4 -3/2 -1 2

16. 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= [ ] 7 5 4 7 -1 -3

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