Let \(a\in \mathrm{R}\) and \(A\) be a matrix of order \(3\times 3\) such that \(\det (A)=-4\) and \(A+I=\left[\begin{ma…

Let \(a\in \mathrm{R}\) and \(A\) be a matrix of order \(3\times 3\) such that \(\det (A)=-4\) and \(A+I=\left[\begin{matrix}1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2\end{matrix}\right]\), where \(I\) is the identity matrix of order \(3\times 3\). If \(\det ((\mathrm{a}+1)adj((\mathrm{a})\mathrm{A}))\) is \({2}^{\mathrm{m}}{3}^{\mathrm{n}},\mathrm{m},\mathrm{n}\in\) \({0,1,2,\ldots ..20}\), then \(\mathrm{m}+\mathrm{n}\) is equal to :

[JEE Main 2025, 2 Apr (Shift 1)]