This document discusses inverse matrices and provides examples of finding the inverse of matrices using two methods: 1) Solving simultaneous equations: Forming equations by setting the product of the matrix and its potential inverse equal to the identity matrix. 2) Using a formula: For 2x2 matrices, the inverse is determined by calculating the determinant of the original matrix and arranging the cofactors. Examples are provided to demonstrate both methods. Exercises at the end ask the reader to find specific inverses using each technique.

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 A is the inverse of matrix B. 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=Bˉ¹. B is the inverse matrix of A, B=Aˉ¹.

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 = - d = Substitute c = - in equation 1 3a + (- ) = 1 a = Substitute d = in equation 3 3b + ( ) = 0 b = - Therefore, Aˉ¹=[ ] Check the answer; AAˉ¹= [ ][ ] = [ ]= I 1 3 4 0 0 1

12. Example 1 Given the matrix B, find the inverse Bˉ¹by using the method of solving simultaneous linear equations. B= [ ] Solution: Let Bˉ¹= [ ] [][ ]=[ ] [ ]=[ ] 4e + 3g = 1 1 4f + 4h = 0 3 4e + 4g = 0 2 4f + 4h = 1 4 3 4 4 e f g h 0 0 1 3 4 4 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 = - Therefore, Bˉ¹= [ ]

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 Aˉ¹ = [ ] [ ] ad-bc is the determinant and written as |A| a b c d d -b -c a

15. Example 2 Find the inverse of the , by using the formula a) G =[] Determinant, |G|= ad – bc = (4x2)-(3x2) = 2 Therefore, Gˉ¹=[ ] = [ ] 3 2 2 -3 -2 4

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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