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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
  2. Serial order wise
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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 = 𝟏﷮𝟕﷯ (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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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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