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

For some $\alpha, \beta \in \mathbf{R}$, let $A=\left[\begin{array}{ll}\alpha & 2 \\ 1 & 2\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 1 \\ 1 & \beta\end{array}\right]$ be such that $A^2-4 A+2 I=B^2-3 B+I=O$. Then $\left(\operatorname{det}\left(\operatorname{adj}\left(A^3-B^3\right)\right)\right)^2$ is equal to $\_\_\_\_$ .

The number of singular matrices of order 2 , whose elements are from the set $\{2,3,6,9\}$, is __________.

Let $A=\left[\begin{array}{ccc}\cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta\end{array}\right]$. If for some $\theta \in(0, \pi), A^2=A^T$, then the sum of the diagonal elements of the matrix $(\mathrm{A}+\mathrm{I})^3+(\mathrm{A}-\mathrm{I})^3-6 \mathrm{~A}$ is equal to _________ .