If $\begin{bmatrix} 2 & 3 & 4 \end{bmatrix} \begin{bmatrix} 1 & x & 3 \\ 2 & 4 & 5 \\ 3 & 2 & x \end{bmatrix} \begin{bmatrix} x \\ 2 \\ 0 \end{bmatrix} = O$,then $x = $ . . . . . .

The matrix $A = \begin{bmatrix} 1 & -3 & -4 \\ -1 & 3 & 4 \\ 1 & -3 & -4 \end{bmatrix}$ is nilpotent of index:

If $A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$,then $A^{100} = $

If $A = \begin{bmatrix} 1 & 2 & -1 \\ -1 & 0 & 2 \\ 1 & 2 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} -3 & -2 & 4 \\ 2 & 2 & -1 \\ -2 & 0 & 3 \end{bmatrix}$, then $A^2 = $

If $A = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 2 & 3 & 0 & 0 \\ 4 & 5 & 6 & 0 \\ 7 & 8 & 9 & 10 \end{bmatrix}$,then $A$ is

Let $A$ be a $2 \times 2$ real matrix with entries from $\{0, 1\}$ and $|A| \neq 0$. Consider the following two statements:
$(P)$ If $A \neq I_{2}$,then $|A| = -1$
$(Q)$ If $|A| = 1$,then $\operatorname{tr}(A) = 2$
where $I_{2}$ denotes the $2 \times 2$ identity matrix and $\operatorname{tr}(A)$ denotes the sum of the diagonal entries of $A$. Then: