Prepared by

Prepared by

• 1.
  4.7 INVERSE MATRIXPreparedBy:Sim Win NeeKeu Pei SanChoon Siang YongCh’ngTjeYiePrepared by:Sim Win NeeKeu Pei SanChoon Siang YongCh’ngTjeYie
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  Inverse matrixThe inverseof a square matrix is another matrix such that when the two are multiplied together, in any order, the product is an identify matrix.
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  Inverse matrixWhen anumber , 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 .
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  Example:Determine whether matrixP is the inverse of matrix Q.a) A=[][] 7 -4-5 345 7 , B=b) A=[] ,B=[]53 853 2
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  Solution:a)AB=[][]45 7 7 -4-5 3= []21+(-20) -12+1235+(-35) -20+21= []00 1
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  BA = [][]7 -4-5 33 45 7= [] 21+(-20) 28+(-28)-15+15 -20+21= []00 1AB= I and BA= ITherefore; A is the inverse matrix of B, A=B-1. B is the inverse matrix of A, B=A-1.
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  b) AB= [][]53 88 53 2 = []16+15 10+1024+24 15+16= [] 2048 31( Not equal to I )Therefore; A is not the inverse matrix of matrix B, A is not equal to B.
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  Exercise:Determine whether thematrix A and B are the inverses of one another.1. A= [], and B= [] 3 -2-4 3 2 3 2. A= [] , and B= []38 5-38 5
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  a. Method ofsolving simultaneous equationsGiven, matrix A = [ ]To find the inverse of matrix A, let A⁻1 =[ ]A x A⁻1 = IThen; [ ][ ]=[ ][ ]=[ ] Equal Matrices13 4 a b c d 00 113 4a bc d3a + c 3b + d 3a + 4c 3b + 4d00 1
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  3a + c= 1 1 3b + d = 0 3 3a + 4c = 0 2 3b + 4d = 1 41-2 : -3c = 1 3-4 : -3d = -1 c = -⅓ d = ⅓Substitute c =-⅓ in equation 1 3a + (-⅓) = 1 a = 4⁄9Substitute d = ⅓ in equation 33b + (⅓) = 0 b = - 1/9Therefore, A⁻1=[ ]Check the answer; AA⁻1 = [ ][ ] = [ ]= I4/9 -1/9-1/3 1/313 44/9 -1/9-1/3 1/300 1
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  Example 1Given thematrix 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 34e + 4g = 0 2 4f + 4h = 1 4 34 4e fg h34 400 1e fg h4e +3g 4f +3h4e +4f 4f +4g00 1
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  2-1 : g = -1 4-3 : h = 1So, 4e + 3(-1) = 1 so, 4f + (1)= 0 e = 1 f = -3⁄4Therefore, B⁻1 = [ ]-3⁄4-1 1
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  B. Using FormulaWecan 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 bc d d -b-c dd/ad – bc -b/ ad-bc-c/ ad-bc a/ad-bc
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  Example 2Find theinverse of the , by using the formula a) G =[]Determinant, |G|= ad – bc = (4x2)-(3x2) = 2Therefore, G⁻1 =1/2 [ ] = [ ]32 2-3-2 4 -3/2-1 2
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  1. Using themethod 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= [ ]75 47-1 -3
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