If the matrix A = 0 2 K - 1 satisfies A A 3 + 3 I = 2 I , then the value of K is

$\mathrm{A}\left({\mathrm{A}}^3+3\mathrm{I}\right)=2\mathrm{I}$

${\mathrm{A}}^2=\left[\begin{array}{cc}0& 2\\ \mathrm{K}& -1\end{array}\right]\left[\begin{array}{cc}0& 2\\ \mathrm{K}& -1\end{array}\right]$

$=\left[\begin{array}{cc}2\mathrm{K}& -2\\ -\mathrm{K}& 2\mathrm{K}+1\end{array}\right]$

${\mathrm{A}}^3=\left[\begin{array}{cc}2\mathrm{K}& -2\\ -\mathrm{K}& 2\mathrm{K}+1\end{array}\right]\left[\begin{array}{cc}0& -1\\ \mathrm{K}& 2\end{array}\right]$

$=\left[\begin{array}{cc}-2\mathrm{K}& 4\mathrm{K}+2\\ 2{\mathrm{K}}^2+\mathrm{K}& -2\mathrm{K}-2\mathrm{K}-1\end{array}\right]$

${\mathrm{A}}^4=\left[\begin{array}{cc}2\mathrm{K}& -2\\ -\mathrm{K}& 2\mathrm{K}+1\end{array}\right]\left[\begin{array}{cc}2\mathrm{K}& -2\\ -\mathrm{K}& 2\mathrm{K}+1\end{array}\right]$

$=\left[\begin{array}{cc}4{\mathrm{K}}^2+2\mathrm{K}& -4\mathrm{K}-4\mathrm{K}-2\\ -2{\mathrm{K}}^2-2{\mathrm{K}}^2-\mathrm{K} & 2\mathrm{K}+4{\mathrm{K}}^2+1+4\mathrm{K}\end{array}\right]$

${\mathrm{A}}^4+3\mathrm{AI}=2\mathrm{I}$

$2\mathrm{I}=\left[\begin{array}{cc}4{\mathrm{K}}^2+2\mathrm{K}& -8\mathrm{K}-2\\ -4{\mathrm{K}}^2-\mathrm{K}& 4{\mathrm{K}}^2+6\mathrm{K}+1\end{array}\right]+3\left[\begin{array}{cc}0& 2\\ \mathrm{K}& -1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$2\mathrm{I}=\left[\begin{array}{cc}4{\mathrm{K}}^2+2\mathrm{K}& -8\mathrm{K}-2\\ -4{\mathrm{K}}^2-\mathrm{K}& 4{\mathrm{K}}^2+6\mathrm{K}+1\end{array}\right]+3\left[\begin{array}{cc}0& 2\\ \mathrm{K}& -1\end{array}\right]$

$2\mathrm{I}=\left[\begin{array}{cc}4{\mathrm{K}}^2+2\mathrm{K}& 6-8\mathrm{K}-2\\ 3\mathrm{K}-\mathrm{K}-4{\mathrm{K}}^2& 4{\mathrm{K}}^2+6\mathrm{K}+1-3\end{array}\right]$

$4-8\mathrm{K}=0$

$\mathrm{K}=\frac{1}{2}$