If $A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$,then the incorrect option among the following is

If $A = \begin{bmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{bmatrix}$,then find the matrix $X$ such that $2A + 3X = 5B$.

If $A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix}$,then $A^n = 2^k A$,where $k = $

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:

For the matrix $A = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$,find the values of $a$ and $b$ such that $A^{2} + aA + bI = 0$.

If $A=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]$,$P=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $X=A P A^T$,then $A^T X^{50} A=$