Let . If det (adj (adj (3A)) = , m, n N, then m + n is equal to [2025]

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Solution

Q.
Let $A = [\begin{matrix} 2 & 2 + p & 2 + p + q \\ 4 & 6 + 2 p & 8 + 3 p + 2 q \\ 6 & 12 + 3 p & 20 + 6 p + 3 q \end{matrix}]$ . If det (adj (adj (3A))) = $2^{m} \cdot 3^{n}$ , m, n $\in$ N, then m + n is equal to [2025]

1 20

2 24

3 26

4 22

Ans.
(2)

$A = [\begin{matrix} 2 & 2 + p & 2 + p + q \\ 4 & 6 + 2 p & 8 + 3 p + 2 q \\ 6 & 12 + 3 p & 20 + 6 p + 3 q \end{matrix}]$

$R_{2} \to R_{2} - 2 R_{1} and R_{3} \to R_{3} - 3 R_{1}$

$A = [\begin{matrix} 2 & 2 + p & 2 + p + q \\ 0 & 2 & 4 + p \\ 0 & 6 & 14 + 3 p \end{matrix}]$

$∴ | A | = 2 (28 + 6 p - 24 - 6 p) = 8 = 2^{3}$

$∴ | adj (adj (3 A)) | = | 3 A |^{{(3 - 1)}^{2}}$

$= | 3 A |^{4} = {(3^{3} | A |)}^{4} = 3^{12} {(2^{3})}^{4} = 2^{12} 3^{12}$

$∴ m = n = 12 \Rightarrow m + n = 24$ .

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