Let A = [ 2 1 0 1 2 - 1 0 - 1 2 ] . If | adj ( adj ( adj ( 2 A ) ) ) | = ( 16 ) n , then n is equal to [2023]

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Solution

Q.
Let $A =$ $[\begin{matrix} 2 & 1 & 0 \\ 1 & 2 & - 1 \\ 0 & - 1 & 2 \end{matrix}]$ . If $| adj (adj (adj (2 A))) |$ $= {(16)}^{n},$ then $n$ is equal to [2023]

1 9

2 10

3 12

4 8

Ans.
(2)

We have,
$A = [\begin{matrix} 2 & 1 & 0 \\ 1 & 2 & - 1 \\ 0 & - 1 & 2 \end{matrix}] \Rightarrow | A | = 4$

Now, $| adj (adj (adj (2 A))) | = | 2 A |^{{(n - 1)}^{3}} = | 2 A |^{2^{3}} = | 2 A |^{8}$

$= {[2^{3} | A |]}^{8} = 8^{8} \cdot 4^{8} = 32^{8} = 2^{8} \cdot 16^{8} = 16^{2} \cdot 16^{8} = 16^{n}$

$\Rightarrow n = 10$

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