Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = 3\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$. Then the maximum value of $\operatorname{det}(A)$ is:

Let $M$ and $N$ be two invertible square matrices over $\mathbb{R}$ of order $2$ such that $N$ is diagonal. Then $M N M^{-1}$ is diagonal . . . . . .

If $x, y$ are any two non-zero real numbers, $a_{i j} = xi + yj$, $A = \{a_{i j}\}_{n \times n}$ and $P, Q$ are two $n \times n$ matrices such that $A = xP + yQ$, then

If $A=\left[\begin{array}{ccc}1 & 2 & 3 \\ 1 & 1 & 1 \\ 1 & -1 & 1\end{array}\right], B=\left[\begin{array}{lll}1 & 1 & 0 \\ 0 & 1 & 3 \\ 3 & 0 & 4\end{array}\right]$,and $C=\left[\begin{array}{lll}2 & 0 & 1 \\ 0 & 1 & 0 \\ 3 & 2 & 1\end{array}\right]$,then $\left(\left(\left((A B C)^{-1}\right)^T\right)^{-1}\right)^T=$

If matrix $A = [a_{ij}]_{3 \times 3}$ and $B = [b_{ij}]_{3 \times 3}$,where $a_{ij} + a_{ji} = 0$ and $b_{ij} - b_{ji} = 0$ for all $i, j$,then $A^4B^3$ is:

Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. $\operatorname{Tr}(A)$ denotes the sum of diagonal entries of $A$. Assume that $A^2=I$.
Statement $I$: If $A \neq I$ and $A \neq -I$,then $\operatorname{det}(A) = -1$.
Statement $II$: If $A \neq I$ and $A \neq -I$,then $\operatorname{Tr}(A) \neq 0$.