If $A$ and $B$ are two matrices such that $AB = B$ and $BA = A$,then $A^{2} + B^{2}$ equals

If $A = \begin{bmatrix} 2 & 3 \\ 0 & -1 \end{bmatrix}$,then the value of $\det(A^4) + \det(A^{10} - (\operatorname{adj}(2A))^{10})$ is equal to ........

Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $\begin{bmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1 \end{bmatrix}$ where each of $a, b$,and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is

For a square matrix $B$ of order $3$,if $B^T=B^{-1}$ and $|B|=1$,then $|B-I|=$

Let $A = \begin{bmatrix} p & 13 \\ -13 & p \end{bmatrix}$ and $B = \begin{bmatrix} 4q & 85 \\ -2 & 1 \end{bmatrix}$ where $p, q \in N$. It is given that $|A| = |B|$ and $p, q \in [1, 1000]$. Then the total number of ordered pairs $(p, q)$ is:

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