Let A be a 3 × 3 matrix such that A + A^T = O. If A [1; -1; 0] = [3; 3; 2], A^2 [1; -1; 0] = [-3; 19; -24] and det ( adj (2 adj ( A + I )))=(2)^α…

Let A be a $\displaystyle 3 \times 3$ matrix such that $\displaystyle \mathrm{A}+\mathrm{A}^{\mathrm{T}}=\mathrm{O}$. If $\displaystyle \mathrm{A}\left[\begin{array}{c}1 \\ -1 \\ 0\end{array}\right]=\left[\begin{array}{l}3 \\ 3 \\ 2\end{array}\right], \mathrm{A}^2\left[\begin{array}{c}1 \\ -1 \\ 0\end{array}\right]=\left[\begin{array}{c}-3 \\ 19 \\ -24\end{array}\right]$ and $\displaystyle \operatorname{det}(\operatorname{adj}(2 \operatorname{adj}(\mathrm{~A}+\mathrm{I})))=(2)^\alpha \cdot(3)^\beta \cdot(11)^\gamma, \alpha, \beta, \gamma$ are non-negative integers, then $\displaystyle \alpha+\beta+\gamma$ is equal to $\displaystyle \_\_\_\_$