Let a ∈ R and A be a matrix of order 3 × 3 such that det ( A )=-4 and A + I = [1, a, 1; 2, 1, 0; a, 1, 2], where I is the identity matrix of order 3…

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