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 _________ .

Let $I$ be the identity matrix of order $3 \times 3$ and for the matrix $A=\left[\begin{array}{ccc}\lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2\end{array}\right],|A|=-1$. Let $B$ be the inverse of the matrix $\operatorname{adj}\left(\operatorname{Aadj}\left(A^2\right)\right)$. Then $|(\lambda \mathrm{B}+\mathrm{I})|$ is equal to______

Let $S=\left\{m \in \mathbf{Z}: A^{m^2}+A^m=3 I-A^{-6}\right\}$, where $A=\left[\begin{array}{cc}2 & -1 \\ 1 & 0\end{array}\right]$. Then $n(S)$ is equal to __________.