Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$. Then the maximum value of $\operatorname{det}(\mathrm{A})$ is……………………

Find $\mathrm{AB},$ if $\mathrm{A}=\left[\begin{array}{rr}0 & -1 \\ 0 & 2\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}3 & 5 \\ 0 & 0\end{array}\right]$.

Let the determinant of a $3 \times 3$ matrix A be 6 then $B$  is a matrix defined by $B = 5{A^2}$. Then det of $B$  is

If $A = \left[ {\begin{array}{*{20}{c}}0&i\\{ – i}&0\end{array}} \right]$, then the value of ${A^{40}}$ is

Let $S=\left\{\left(\begin{array}{cc}-1 & a \\ 0 & b\end{array}\right) ; a, b \in\{1,2,3, \ldots 100\}\right\}$ and let $T_{n}=\left\{A \in S: A^{n(n+1)}=I\right\}$. Then the number of elements in $\bigcap \limits_{n=1}^{100} T_{n}$ is