Mathematics - Matrices Question with Solution | TestHub

$\begin{array}{rl}& |3\mathrm{adj}(2\mathrm{adj}(|A|A))|=\mid 3\mathrm{adj}(2|A{|}^2\mathrm{adj}(A)\mid \\ & ={\left|3.2^2\right|A|}^4\mathrm{adj}({\mathrm{adj}(A)\left|=2^6{3}^3\right|A|}^{12}|A{|}^4\\ & =2^6{3}^3|A{|}^{16}=2^{-10}{3}^{-13}\\ & \Rightarrow |A{|}^{16}=2^{-16}{3}^{-16}\Rightarrow |A|=2^{-1}{3}^{-1}\end{array}$ $\begin{array}{rl}& \text{ Now }|3\mathrm{adj}(2\mathrm{A})|=\left|3.2^2\mathrm{adj}(\mathrm{A})\right|\\ & =2^6{3}^3|\mathrm{A}{|}^2=2^{-\mathrm{m}}{3}^{-\mathrm{n}}\\ & \Rightarrow 2^6{3}^3{2}^{-2}{3}^{-2}=2^{-\mathrm{m}}{3}^{-\mathrm{n}}\\ & \Rightarrow 2^{-\mathrm{m}}{3}^{-\mathrm{n}}=2^4{3}^1\\ & \Rightarrow \mathrm{m}=-4,\mathrm{n}=-1\\ & \Rightarrow |3\mathrm{m}+2\mathrm{n}|=|-12-2|=14\end{array}$