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Multiplication of Matrices

Suppose a Mat...

Question

Suppose a Matrix A satisfies $A^{2} - 5 A + 7 I = 0$ If $A^{5} = a A + b I$ , then the values of 2a + b is.

-87

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-105

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1453

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1155

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Solution

The correct option is A -87
$A^{3} = A A^{2} = A (5 A - 7 I) = 5 A^{2} - 7 A = 5 (5 A - 7 I) - 7 A = 18 A - 35 I A^{4} = A . A^{3} = A (18 A - 35 I) = 18 (5 A - 7 I) - 35 A A^{5} = 149 A - 385 I = 55 A - 126 I A^{5} = 149 A - 385 I a = 149, b = - 385$

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A^{2} - 5 A + 7 I = 0. I f A^{5} = a A + b I

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