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If A = 1 0 0 0 1 1 0 - 2 4 , I = 1 0 0 0 1 0 0 0 1 and A - 1 = 1 6 A 2 + c A + d then the sum of values of c and d is

Correct answer

5

Step-by-step solution

Method I We evaluate A 2 and A 3 and write the given equation as A A – 1 = I = 1 6 A 3 + c A 2 + d A . Comparing the corresponding elements on both the sides, we get c = - 6 , d = 11 . Method I I Characteristic equation for A is A - x I = 0 = 1 - x 0 0 0 1 - x 1 0 - 2 4 - x = 0 ⇒ 1 - x 1 - x 4 - x + 2 = 0 Hence characteristic equation is x 3 - 6 x 2 + 11 x - 6 = 0 Then by Caley Hamilton theorem A 3 - 6 A 2 + 11 A - 6 I = 0 multiply by A - 1 both the sides, we get 1 6 A 2 - 6 A + 11 I = A - 1 ...(i) given A - 1 = 1

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