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 $i = \sqrt{-1}$)

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

If $a > 0$ and the discriminant of $ax^2 + 2bx + c$ is negative,then $\left| \begin{array}{ccc} a & b & ax + b \\ b & c & bx + c \\ ax + b & bx + c & 0 \end{array} \right|$ is

Let $M = \begin{bmatrix} a & b \\ b & c \end{bmatrix}$ be a $2 \times 2$ symmetric matrix with integer entries $a, b, c$. Which of the following conditions guarantee that $M$ is invertible?
$(A)$ The first column equals the transpose of the second row.
$(B)$ The second row equals the transpose of the first column.
$(C)$ $M$ is a diagonal matrix with non-zero diagonal entries.
$(D)$ $ac$ is not a perfect square.

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$, then $A^3 - 4A^2 - 6A$ is equal to:

Let $X = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ and $A = \begin{bmatrix} -1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1 \end{bmatrix}$. For $k \in N$,if $X^{T} A^{k} X = 33$,then $k$ is equal to: