Let $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$,$\alpha \in R$ such that $A^{32} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. Then a value of $\alpha$ is

Let $S$ be the set containing all $3 \times 3$ matrices with entries from $\{-1, 0, 1\}$. The total number of matrices $A \in S$ such that the sum of all the diagonal elements of $A^{T}A$ is $6$ is:

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2+x-1=0$. Consider the set $T=\{1, \alpha, \beta\}$. For a $3 \times 3$ matrix $M=(a_{ij})$,define $R_i=a_{i1}+a_{i2}+a_{i3}$ and $C_j=a_{1j}+a_{2j}+a_{3j}$ for $i=1,2,3$ and $j=1,2,3$. Match each entry in $List-I$ to the correct entry in $List-II$.

$List-I$	$List-II$
$(P)$ The number of matrices $M=(a_{ij})_{3 \times 3}$ with all entries in $T$ such that $R_i=C_j=0$ for all $i, j$ is	$(1)$ $1$
$(Q)$ The number of symmetric matrices $M=(a_{ij})_{3 \times 3}$ with all entries in $T$ such that $C_j=0$ for all $j$ is	$(2)$ $2$
$(R)$ Let $M=(a_{ij})_{3 \times 3}$ be a skew-symmetric matrix such that $a_{ij} \in T$ for $i>j$. Then the number of elements in the set $\{\begin{bmatrix} x \\ y \\ z \end{bmatrix}: x, y, z \in \mathbb{R}, M\begin{bmatrix} x \\ y \\ z \end{bmatrix}=\begin{bmatrix} a_{12} \\ 0 \\ -a_{23} \end{bmatrix}\}$ is	$(3)$ $\text{Infinite}$
$(S)$ Let $M=(a_{ij})_{3 \times 3}$ be a matrix with all entries in $T$ such that $R_i=0$ for all $i$. Then the absolute value of the determinant of $M$ is	$(4)$ $6$
	$(5)$ $0$

For $M=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ and for any $n \in N$,the matrix $M^{n+1}-M^n=$

Let $M$ be any $3 \times 3$ matrix with entries from the set $\{0, 1, 2\}$. The maximum number of such matrices,for which the sum of diagonal elements of $M^{T}M$ is $7$,is .............

If $z = \begin{bmatrix} 1 & 1+2i & -5i \\ 1-2i & -3 & 5+3i \\ 5i & 5-3i & 7 \end{bmatrix}$, then which of the following is true? (where $i = \sqrt{-1}$)