Let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $P = [p_{ij}]$ be a $2 \times 2$ matrix with $p_{ij} = \omega^{i+j}$. For $P^2 \neq 0$,if $P^k = P$,then $k$ is equal to

Let $|A|=6$ where $A$ is a $3 \times 3$ matrix. If $|adj(3adj(A^{2} \cdot adj(2A)))|=2^{m} \cdot 3^{n}$, $m, n \in N$, then $m+n$ is equal to:

Let $A$ and $B$ be two non-singular matrices of order $3$ such that $A + B = I$ and $A^{-1} + B^{-1} = 2I$. Then $|adj(4AB)|$ is equal to (where $adj(A)$ is the adjoint of matrix $A$):

Let $R = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ be a non-zero $3 \times 3$ matrix,where $x \sin \theta = y \sin \left(\theta + \frac{2 \pi}{3}\right) = z \sin \left(\theta + \frac{4 \pi}{3}\right) \neq 0$,$\theta \in (0, 2 \pi)$. For a square matrix $M$,let $\text{trace}(M)$ denote the sum of all the diagonal entries of $M$. Then,among the statements:
$(I) \text{ Trace}(R) = 0$
$(II) \text{ If trace}(\text{adj}(\text{adj}(R))) = 0, \text{ then } R \text{ has exactly one non-zero entry.}$

If $A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}$, then the determinant of the matrix $(A^{2025} - 3A^{2024} + A^{2023})$ is

Let $A = [a_{ij}]$ be a $2 \times 2$ matrix such that $a_{ij} \in \{0, 1\}$ for all $i$ and $j$. Let the random variable $X$ denote the possible values of the determinant of the matrix $A$. Then,the variance of $X$ is: