For two $3 \times 3$ matrices $A$ and $B$,let $A + B = 2B^T$ and $3A + 2B = I_3$,where $B^T$ is the transpose of $B$ and $I_3$ is the $3 \times 3$ identity matrix. Then:

Compute the following: $\begin{bmatrix} -1 & 4 & -6 \\ 8 & 5 & 16 \\ 2 & 8 & 5 \end{bmatrix} + \begin{bmatrix} 12 & 7 & 6 \\ 8 & 0 & 5 \\ 3 & 2 & 4 \end{bmatrix}$

If $A, B, C$ are three $n \times n$ matrices,then $(ABC)' = $

If $A$ and $B$ are symmetric matrices of the same order,then which one of the following is not true?

If $A = \begin{bmatrix} 1 & -2 & 1 \\ 2 & 1 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 1 \\ 3 & 2 \\ 1 & 1 \end{bmatrix}$,then $(AB)^T$ is equal to

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: