Mathematics Question from JEE Main 2025 · 2 April, Shift 1

Let $a \in \mathbb{R}$ and $A$ be a matrix of order $3 \times 3$ such that

$$\det(A) = -4 \quad \text{and} \quad A + I = \left[ \begin{array}{ccc} 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 \end{array} \right]$$

where $I$ is the identity matrix of order $3$. If $\det\big((a+1)\,\mathrm{adj}((a-1)A)\big)$ is $2^m 3^n$, $m, n \in \{0, 1, 2, \ldots, 20\}$, then $m + n$ is equal to: