Mathematics - Matrix Question with Solution | TestHub

$\begin{array}{rl}& |\mathrm{adj}(\mathrm{adj})(\mathrm{adjA})|=81\\ & \Rightarrow |\mathrm{adjA}{|}^4=81\\ & \Rightarrow |\mathrm{adjA}|=3\\ & \Rightarrow |\mathrm{A}{|}^2=3\\ & \Rightarrow |\mathrm{A}|=\sqrt{3}\\ & {\left(|\mathrm{A}{|}^4\right)}^{\frac{(\mathrm{n}-1{)}^2}{2}}=|\mathrm{A}{|}^{{\mathrm{n}}^2-5\mathrm{n}-4}\\ & \Rightarrow 2(\mathrm{n}-1{)}^2=3{\mathrm{n}}^2-5\mathrm{n}-4\\ & \Rightarrow 2{\mathrm{n}}^2-4\mathrm{n}+2=3{\mathrm{n}}^2-5\mathrm{n}-4\\ & \Rightarrow {\mathrm{n}}^2-\mathrm{n}-6=0\\ & \Rightarrow (\mathrm{n}-3)(\mathrm{n}+2)=0\\ & \Rightarrow \mathrm{n}=3,-2\\ & \sum _{\mathrm{n}\in \mathrm{s}}\left|{\mathrm{A}}^{{\mathrm{n}}^2+\mathrm{n}}\right|\\ & =\left|{\mathrm{A}}^2\right|+\left|{\mathrm{A}}^{12}\right|\\ & =3+36=3+729=732\end{array}$