Let A be a non-singular matrix of order 3 . If det(3 adj(2 adj(( det A) A)))=3^-13 · 2^-10 and det(3 adj(2 ~A))=2^ m · 3^ n, then |3 ~m+2 n| is equal…

$\begin{aligned} & |3 \operatorname{adj}(2 \operatorname{adj}(|A| A))|=\mid 3 \operatorname{adj}\left(2|A|^2 \operatorname{adj}(A) \mid\right. \\ & =\left.\left|3.2^2\right| A\right|^4 \operatorname{adj}\left(\left.\operatorname{adj}(A)\left|=2^6 3^3\right| A\right|^{12}|A|^4\right. \\ & =2^6 3^3|A|^{16}=2^{-10} 3^{-13} \\ & \Rightarrow|A|^{16}=2^{-16} 3^{-16} \Rightarrow|A|=2^{-1} 3^{-1}\end{aligned}$​∣3adj(2adj(∣A∣A))∣=∣3adj(2∣A∣2adj(A)∣=​3.22​A​4adj(adj(A)​=2633​A​12∣A∣4=2633∣A∣16=2−103−13⇒∣A∣16=2−163−16⇒∣A∣=2−13−1​ $\begin{aligned} & \text { Now }|3 \operatorname{adj}(2 \mathrm{~A})|=\left|3.2^2 \operatorname{adj}(\mathrm{A})\right| \\ & =2^6 3^3|\mathrm{~A}|^2=2^{-\mathrm{m}} 3^{-\mathrm{n}} \\ & \Rightarrow 2^6 3^3 2^{-2} 3^{-2}=2^{-\mathrm{m}} 3^{-\mathrm{n}} \\ & \Rightarrow 2^{-\mathrm{m}} 3^{-\mathrm{n}}=2^4 3^1 \\ & \Rightarrow \mathrm{m}=-4, \mathrm{n}=-1 \\ & \Rightarrow|3 \mathrm{~m}+2 \mathrm{n}|=|-12-2|=14\end{aligned}$​ Now ∣3adj(2 A)∣=​3.22adj(A)​=2633∣ A∣2=2−m3−n⇒26332−23−2=2−m3−n⇒2−m3−n=2431⇒m=−4,n=−1⇒∣3 m+2n∣=∣−12−2∣=14​