First slide

Operations of matrix

Question

Suppose matrix $A$ satisfies the equation $A^{2} - 5 A + 7 I = O .$ If $A^{8} = a A + b I,$ then value of $a$ is

Moderate

A

1265
B

2599
C

-2599
D

0

Solution

$\begin{array}{l} A^{2} = 5 A - 7 I \\ \Rightarrow A^{4} = {(A^{2})}^{2} = (5 A - 7 I) (5 A - 7 I) \\ = 25 A^{2} - 70 A + 49 I \\ = 25 (5 A - 7 I) - 70 A + 49 I \\ = 55 A - 126 I \\ \Rightarrow A^{8} = {(A^{4})}^{2} = (55 A - 126 I) (55 A - 126 I) \\ = 3025 A^{2} - 13860 A + 15876 I \\ = 3025 (5 A - 7 I) - 13860 A + 158766 I \\ = 1265 A - 5299 I \end{array}$
Thus, $a = 1265 .$

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