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Addition and Subtraction of a Matrix

If A2 = 3A -2...

Question

If $A^{2} = 3 A - 2 I$ and $A^{8} = a A + b I$ , then $a + b$ is

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Solution

The correct option is C 1
We have, $A^{2} = 3 A - 2 I \Rightarrow A^{3} = 3 A^{2} - 2 A = 3 (3 A - 2 I) - 2 A$
$\Rightarrow A^{3} = 7 A - 6 I$
$\Rightarrow A^{4} = 7 A^{2} - 6 A$
$\Rightarrow A^{4} = 7 (3 A - 2 I) - 6 A$
$\Rightarrow A^{4} = 15 A - 14 I$

So $A^{n} = (2^{n} - 1) A - (2^{n} - 2) I$

So $A^{8} = 255 A - 254 I$

$∴ a + b = 1$

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