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

Matrix $A_r = \begin{bmatrix} r & r-1 \\ r-1 & r \end{bmatrix}$ for $r = 1, 2, 3, \dots$. If $\sum_{r=1}^{109} |A_r| = (\sqrt{10})^k$,then $k = $ . . . . . . . Where $|A_r| = \det(A_r)$.

For $3 \times 3$ matrices $M$ and $N$,which of the following statement$(s)$ is (are) $NOT$ correct?

Let $A, B, C$ be $3 \times 3$ non-singular matrices and $I$ be the identity matrix of order three. If $A B A = B A^2 B$ and $A^3 = I$, then $A B^4 - B^4 A = $

Let the minimum $m$ $(m \in Z^+)$ be defined as the power of a square matrix $A$ such that $A^m = I$. If $A^5 = I$ and $ABA^{-1} = B^2$,then the power of matrix $B$ such that $B^k = I$ is between:

If $A$ and $B$ are $3 \times 3$ matrices such that $AB = A$ and $BA = B$,then