Suppose a matrix $A$ satisfies $A^{2} - 5 A + 7 I = 0$ . If $A^{8} = a A + b I$ , find $a$ .

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Solution

Given $A^{2} - 5 A + 7 I = 0$
$A^{2} = 5 A - 7 I$
$A^{4} = (5 A - 7 I)^{2}$
$= 25 A^{2} - 70 A + 49 I$
$= 25 (5 A - 7 I) - 70 A + 49 I$
$\Rightarrow A^{4} = 55 A - 126 I$
So, $A^{8} = 3025 A^{2} - 13860 A + 15876 I$
$= 3025 (5 A - 7 I) - 13860 A + 15876 I$
$= 1265 A - 5299 I$
Comparing with $A^{8} = a A + b I$ (given)
$\Rightarrow a = 1265$

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