If A = 1 5 6 7 [ 5 6 7 6 7 8 7 8 9 ] , then | adj ( adj ( 2 A ) ) | is equal to [2023]

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Solution

Q.
If $A = \frac{1}{5! 6! 7!} [\begin{matrix} 5! & 6! & 7! \\ 6! & 7! & 8! \\ 7! & 8! & 9! \end{matrix}],$ then $| adj (adj (2 A)) |$ is equal to [2023]

1 $2^{12}$

2 $2^{20}$

3 $2^{8}$

4 $2^{16}$

Ans.
(4)

Given $A = \frac{1}{5! 6! 7!} [\begin{matrix} 5! & 6! & 7! \\ 6! & 7! & 8! \\ 7! & 8! & 9! \end{matrix}]$

$| A |$ $= \frac{1}{5! 6! 7!} 5! 6! 7!$ $| \begin{matrix} 1 & 6 & 42 \\ 1 & 7 & 56 \\ 1 & 8 & 72 \end{matrix} |$

Applying $R_{2} \to R_{2} - R_{1}$ and $R_{3} \to R_{3} - R_{2}$ , we get

$| A |$ $=$ $| \begin{matrix} 1 & 6 & 42 \\ 0 & 1 & 14 \\ 0 & 1 & 16 \end{matrix} |$

Now, $| adj (adj (2 A)) |$ $= | 2 A |^{{(n - 1)}^{2}} = | 2 A |^{4} = {(2^{3} | A |)}^{4} = 2^{12} | A |^{4} = 2^{12} 2^{4} = 2^{16}$

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