If $A = \begin{bmatrix} 0 & 2 & 0 \\ 0 & 0 & 3 \\ -2 & 2 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 5 & -4 & 0 \end{bmatrix}$,then the element of the $3^{rd}$ row and $3^{rd}$ column in $AB$ will be:

If $P = \begin{bmatrix} 1 & 2 & 1 \\ 1 & 3 & 1 \end{bmatrix}$ and $Q = P P^{T}$, then the value of the determinant of $Q$ 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:

If $A$ and $B$ are square matrices of order $n$ such that $A^{2}-B^{2}=(A-B)(A+B)$,then which of the following will be true?

For $A = \begin{bmatrix} 0 & 0 & 3 \\ 0 & 3 & 0 \\ 3 & 0 & 0 \end{bmatrix}$,which statement is correct?

If $A=\left[\begin{array}{cccc}2 & 1 & 3 & -1 \\ 1 & -2 & 2 & -3\end{array}\right]$,$B=\left[\begin{array}{cccc}2 & 1 & 0 & 3 \\ 1 & -1 & 2 & 3\end{array}\right]$ and $2A+3B-5C=0$,then $C=$