$\text{ If the matrix }\mathrm{A}=\left(\begin{array}{cc}0& 2\\ k& -1\end{array}\right)\text{ satisfies }\mathrm{A}\left({\mathrm{A}}^3+3\mathrm{I}\right)=2\mathrm{I}\text{, then the value of }\mathrm{k}\text{ is : }$

$$\begin{gathered} A=\left[\begin{array}{cc}0& 2\\ K& -1\end{array}\right] \\ \begin{array}{r}{\mathrm{A}}^4+3\mathrm{IA}=2\mathrm{I}\\ \Rightarrow {\mathrm{A}}^4=2\mathrm{I}-3\mathrm{A} \\ \text{ Also characteristic equation of }\mathrm{A}\text{ is }\end{array} \\ \begin{array}{l}\mathrm{IA}-\lambda \mathrm{II}=0\\ \Rightarrow \left|\begin{array}{cc}0-\lambda & 2\\ \mathrm{k}& -1-\lambda \end{array}\right|=0\\ \Rightarrow \lambda +{\lambda }^2-2\mathrm{k}=0\\ \Rightarrow \mathrm{A}+{\mathrm{A}}^2=2\mathrm{K}.\mathrm{I}\\ \Rightarrow {\mathrm{A}}^2=2\mathrm{KI}-\mathrm{A}\\ \Rightarrow {\mathrm{A}}^4=4{\mathrm{K}}^2\mathrm{I}+{\mathrm{A}}^2-4\mathrm{AK}\\ \text{ Put }{\mathrm{A}}^2=2\mathrm{KI}-\mathrm{A}\\ \text{ and }{\mathrm{A}}^4=2\mathrm{I}-3\mathrm{A}\\ 2\mathrm{I}-3\mathrm{A}=4{\mathrm{K}}^2\mathrm{I}+2\mathrm{KI}-\mathrm{A}-4\mathrm{AK}\\ \Rightarrow \mathrm{I}\left(2-2\mathrm{K}-4{\mathrm{K}}^2\right)=\mathrm{A}(2-4\mathrm{K})\\ \Rightarrow -2\mathrm{I}\left(2{\mathrm{K}}^2+\mathrm{K}-1\right)=2\mathrm{A}(1-2\mathrm{K})\\ \Rightarrow -2\mathrm{I}(2\mathrm{K}-1)(\mathrm{K}+1)=2\mathrm{A}(1-2\mathrm{K})\\ \Rightarrow (2\mathrm{K}-1)\left(2\mathrm{A}\right)-2\mathrm{I}(2\mathrm{K}-1)(\mathrm{K}+1)=0\\ \Rightarrow (2\mathrm{K}-1)[2\mathrm{A}-2\mathrm{I}(\mathrm{K}+1\left)\right]=0\\ \Rightarrow \mathrm{K}=\frac{1}{2}\end{array} \end{gathered}$$