The number of integers $x$ satisfying $-3 x^4 + \operatorname{det}\begin{bmatrix} 1 & x & x^2 \\ 1 & x^2 & x^4 \\ 1 & x^3 & x^6 \end{bmatrix} = 0$ is equal to

Match the items of List-$I$ with those of List-$II$. The correct match is:

If $A$,$B$,and $C$ are square matrices of order $3$ such that $A = \begin{bmatrix} x & 0 & 1 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ and $|B| = 36$,$|C| = 4$,$(x, y, z \in \mathbb{N})$ and $|ABC| = 1152$,then the minimum value of $x + y + z$ is

Let $A = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$,$B = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$,and $P = \begin{bmatrix} 0 & 1 & 0 \\ x & 0 & 0 \\ 0 & 0 & y \end{bmatrix}$ be an orthogonal matrix such that $B = PAP^{-1}$ holds. Then:

If $A$ and $B$ are two non-zero $n \times n$ matrices such that $A^2 + B = A^2 B$,then:

Let $S = \{ m \in \mathbb{Z} : A^{m^2} + A^m = 3I - A^{-6} \}$,where $A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}$. Then $n(S)$ is equal to