Let $A$ be a non-singular matrix of order 3 . If $\operatorname{det}(3 \operatorname{adj}(2 \operatorname{adj}((\operatorname{det} A) A)))=3^{-13} \cdot 2^{-10}$ and $\operatorname{det}(3 \operatorname{adj}(2 \mathrm{~A}))=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}$, then $|3 \mathrm{~m}+2 \mathrm{n}|$…

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Let $A$ be a non-singular matrix of order 3 . If $\operatorname{det}(3 \operatorname{adj}(2 \operatorname{adj}((\operatorname{det} A) A)))=3^{-13} \cdot 2^{-10}$ and $\operatorname{det}(3 \operatorname{adj}(2 \mathrm{~A}))=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}$, then $|3 \mathrm{~m}+2 \mathrm{n}|$ is equal to $\qquad$

Aicharya · Accepted Answer

$\begin{aligned} & |3 \operatorname{adj}(2 \operatorname{adj}(|A| A))|=\mid 3 \operatorname{adj}\left(2|A|^2 \operatorname{adj}(A) \midight. \ & =\left.\left|3.2^2ight| Aight|^4 \operatorname{adj}\left(\left.\operatorname{adj}(A)\left|=2^6 3^3ight| Aight|^{12}|A|^4ight. \ & =2^6 3^3|A|^{16}=2^{-10} 3^{-13} \ & \Rightarrow|A|^{16}=2^{-16} 3^{-16} \Rightarrow|A|=2^{-1} 3^{-1}\end{aligned}$ $\begin{aligned} & 	ext { Now }|3 \operatorname{adj}(2 \mathrm{~A})|=\left|3.2^2 \operatorname{adj}(\mathrm{A})ight| \ & =2^6 3^3|\mathrm{~A}|^2=2^{-\mathrm{m}} 3^{-\mathrm{n}} \ & \Rightarrow 2^6 3^3 2^{-2} 3^{-2}=2^{-\mathrm{m}} 3^{-\mathrm{n}} \ & \Rightarrow 2^{-\mathrm{m}} 3^{-\mathrm{n}}=2^4 3^1 \ & \Rightarrow \mathrm{m}=-4, \mathrm{n}=-1 \ & \Rightarrow|3 \mathrm{~m}+2 \mathrm{n}|=|-12-2|=14\end{aligned}$

Let $A$ be a non-singular matrix of order 3 . If $\operatorname{det}(3 \operatorname{adj}(2…

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