This document discusses methods for finding the inverse of a square matrix. It defines an inverse matrix as another matrix such that when the two are multiplied together, the product is the identity matrix. It then provides two methods for calculating the inverse: 1) Solving simultaneous linear equations by setting the product of the original matrix and inverse matrix equal to the identity matrix. 2) Using a formula for the inverse of a 2x2 matrix where the inverse is determined by the matrix elements and the determinant. Examples are shown of applying both methods to find specific inverse matrices.

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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