Let \(P=\left[{p}_{ij}\right]\)and \(Q=\left[{q}_{ij}\right]\) be two square matrices of order \(3\) such that \({q}_{ij…

\(\left|Q\right|=\left|\begin{matrix}2{\text{p}}_{11} & {2}^{2}{\text{p}}_{12} & {2}^{3}{\text{p}}_{13} \\ {2}^{2}{\text{p}}_{21} & {2}^{3}{\text{p}}_{22} & {2}^{4}{\text{p}}_{23} \\ {2}^{3}{\text{p}}_{31} & {2}^{4}{\text{p}}_{32} & {2}^{5}{\text{p}}_{33}\end{matrix}\right|={2}^{10}\)

\(\Rightarrow {2}^{2}⋅2⋅{2}^{3}\left|\begin{matrix}{\text{p}}_{11} & {\text{p}}_{12} & {\text{p}}_{13} \\ 2{\text{p}}_{21} & 2{\text{p}}_{22} & 2{\text{p}}_{23} \\ {2}^{2}{\text{p}}_{31} & {2}^{2}{\text{p}}_{32} & {2}^{2}{\text{p}}_{33}\end{matrix}\right|={2}^{10}\)

\(\Rightarrow {2}^{9}\left|\begin{matrix}{\text{p}}_{11} & {\text{p}}_{12} & {\text{p}}_{13} \\ {\text{p}}_{21} & {\text{p}}_{22} & {\text{p}}_{23} \\ {\text{p}}_{31} & {\text{p}}_{32} & {\text{p}}_{33}\end{matrix}\right|={2}^{10}\)

\(\Rightarrow |\text{P}|=2\)

\(|\operatorname{adj}(\operatorname{adj}(P))|=|P|^{(n-1)^2}\)

\(=|P{|}^{4}={2}^{4}=16\)