Misc 8 - Show that A2 - 5A + 7I = O, if A = [3 1 -1 2] - Miscellaneous

  1. Chapter 3 Class 12 Matrices
  2. Serial order wise
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Misc. 8 If A = [■8(3&1@−1&2)] , show that A2 – 5A + 7I = O First calculating A2 A2 = A.A A2 = [■8(3&1@−1&2)] [■8(3&1@−1&2)] = [■8(3(3)+1(−1)&3(1)+1(2)@−1(3)+2(−1)&−1(1)+2(2))] = [■8(9−1&3+2@−3−2&−1+4)] = [■8(8&5@−5&3)] ∴ A2 = [■8(8&5@−5&3)] Now calculating A2 – 5A – 7I = [■8(8&5@−5&3)] – 5 [■8(3&1@−1&2)] + 7 [■8(1&0@0&1)] = [■8(8&5@−5&3)] – [■8(5(3)&5(1)@5(−1)&5(2))] + [■8(7(1)&7(0)@7(0)&7(1))] = [■8(8&5@−5&3)] – [■8(15&5@−5&10)] + [■8(7&0@0&7)] = [■8(8−15+7&5−5+0@−5−(−5)&3−10+7)] = [■8(0&0@0&0)] = O = R.H.S. ∴ L.H.S = R.H.S Hence proved

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