Ex 4.5, 13 - Show that A2 - 5A + 7I = O. Hence find A-1 - Finding inverse when Equation of matrice given

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  1. Chapter 4 Class 12 Determinants
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Ex 4.5, 13 If A = 3 1 1 2 show that A2 5A + 7I = O. Hence find A 1. Calculating A2 A2 = A.A = 3 1 1 2 3 1 1 2 = 3 3 +1( 1) 3 1 +1(2) 1 3 +2( 1) 1 1 +2(2) = 9 1 3+2 3 2 1+4 = 8 5 5 3 Taking L.H.S A2 5A + 7I = 8 5 5 3 5 3 1 1 2 + 7 1 0 0 1 = 8 5 5 3 5(3) 5(1) 5( 1) 5(2) + 7(1) 7(0) 7(0) 7(1) = 8 5 5 3 15 5 5 10 + 7 0 0 7 = 8 15+7 5 5+0 5 ( 5)+0 5 10+7 = 8 15+7 5 5+0 5+5+0 5 10+7 = 0 0 0 0 = O Thus, A2 5A + 7I = O Hence proved Finding A 1 A2 5A + 7I = O Pre multiplying A-1 both sides A-1 (A2 5A + 7I ) = A-1 O A-1 . A2 5A-1A + 7A-1 = O A-1 AA 5(A-1 A) + 7A-1 = O (A-1A)A 5 (A-1 A) + 7 (A-1 I) = O IA 5I + 7A-1 = O A 5I + 7 A-1 = 0 7A-1 = 5I A A-1 = (5I A) Putting values A-1 = 1 7 5 1 0 0 1 3 1 1 2 = 1 7 5 0 0 5 3 1 1 2 = 1 7 5 3 0 1 0 ( 1) 5 2 = 1 7 2 1 1 3 Thus, A-1 = 1 7 2 1 1 3

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