Given matrix $A=\begin{bmatrix}0 & 2 \\ k & -1\end{bmatrix}$
$A ^{4}+3 IA =2 I$
$\Rightarrow A ^{4}=2 I -3 A$
Also characteristic equation of $A$ is $|A-\lambda I|=0$
$\Rightarrow \begin{vmatrix}0-\lambda & 2 \\ k & -1-\lambda\end{vmatrix}=0$
$\Rightarrow \lambda+\lambda^{2}-2 K =0$
$\Rightarrow A + A ^{2}=2 K . I$
$\Rightarrow A ^{2}=2 KI - A$
$\Rightarrow A ^{4}=4 K ^{2} I + A ^{2}-4 AK$
Put $A ^{2}=2 KI - A$
and $A ^{4}=2 I -3 A$
$2 I -3 A =4 K ^{2} I +2 KI - A -4 AK$
$\Rightarrow I \left(2-2 K -4 K ^{2}\right)= A (2-4 K )$
$\Rightarrow -2 I \left(2 K ^{2}+ K -1\right)=2 A (1-2 K )$
$\Rightarrow -2 I (2 K -1)( K +1)=2 A (1-2 K )$
$\Rightarrow (2 K -1)(2 A )-2 I (2 K -1)( K +1)=0$
$\Rightarrow (2 K -1)[2 A -2 I ( K +1)]=0$
$\Rightarrow K =\frac{1}{2}$