Let A be a matrix such that $A\left[\begin{array}{cc}1& 2\\ 0& 3\end{array}\right]$ is a scalar matrix and $\left|3\mathrm{A}\right| = 108$. Then $A^2$equals

Let $\mathrm{A} \left[\begin{array}{cc}1& 2\\ 0& 3\end{array}\right]=\left[\begin{array}{cc}\mathrm{k}& 0\\ 0& \mathrm{k}\end{array}\right]$

     $\begin{array}{rcl}\mathrm{A}& =& \left[\begin{array}{cc}\mathrm{k}& 0\\ 0& \mathrm{k}\end{array}\right] {\left[\begin{array}{cc}1& 2\\ 0& 3\end{array}\right]}^{-1}\\ & =& \left[\begin{array}{cc}\mathrm{k}& \raisebox{1ex}{$-2\mathrm{k}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\\ 0& \raisebox{1ex}{$\mathrm{k}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\end{array}\right]\\ \left|\mathrm{A}\right|& =& \frac{{\mathrm{k}}^2}{3}\\ \left|3\mathrm{A}\right|& =& 108\\ 3^2 \left|\mathrm{A}\right|& =& 108\\ 9\times \frac{{\mathrm{k}}^2}{3}& =& 108\\ \mathrm{k}& =& \pm 6\\ \therefore \mathrm{A}& =& \left[\begin{array}{cc}6& -4\\ 0& 2\end{array}\right]\\ \therefore {\mathrm{A}}^2& =& \left[\begin{array}{cc}36& -32\\ 0& 4\end{array}\right]\end{array}$