Suppose a matrix $A$ satisfies $A^{2} - 5 A + 7 I = 0$ if $A^{8} = a A + b I$ then $a =$

1265

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1652

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1526

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1562

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Solution

The correct option is A $1265$
$A^{2} = 5 A - 7 I$
$\Rightarrow A^{3} = 5 A^{2} - 7 A = 5 (5 A - 7 I) - 7 A = 18 A - 35 I$
$\Rightarrow A^{4} = 18 A^{2} - 35 A = 18 (5 A - 7 I) - 35 A = 55 A - 126 I$
$\Rightarrow A^{8} = A^{4} \times A^{4} = (55 A - 126 I) (55 A - 126 I)$
$\Rightarrow A^{8} = 3025 A^{2} - 13860 A + 15876 I$
$\Rightarrow A^{8} = 3025 (5 A - 7 I) - 13860 A + 15876 I$
$\Rightarrow A^{8} = 1265 A - 5299 I = a A + b I$
so, $a = 1265$
and $b = - 5299$
Hence, the answer is $1265.$

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