If the matrix A=[ … ] satisfies A(A^3+3I)=2I, then the value of K is

$$A({A}^{3}+3I)=2I$$A(A3+3I)=2I

$${A}^{2}=[\begin{matrix}0 & 2 \\ K & -1\end{matrix}][\begin{matrix}0 & 2 \\ K & -1\end{matrix}]$$A2=[0K​2−1​][0K​2−1​]

$$=[\begin{matrix}2K & -2 \\ -K & 2K+1\end{matrix}]$$=[2K−K​−22K+1​]

$${A}^{3}=[\begin{matrix}2K & -2 \\ -K & 2K+1\end{matrix}][\begin{matrix}0 & -1 \\ K & 2\end{matrix}]$$A3=[2K−K​−22K+1​][0K​−12​]

$$=[\begin{matrix}-2K & 4K+2 \\ 2{K}^{2}+K & -2K-2K-1\end{matrix}]$$=[−2K2K2+K​4K+2−2K−2K−1​]

$${A}^{4}=[\begin{matrix}2K & -2 \\ -K & 2K+1\end{matrix}][\begin{matrix}2K & -2 \\ -K & 2K+1\end{matrix}]$$A4=[2K−K​−22K+1​][2K−K​−22K+1​]

$$=[\begin{matrix}4{K}^{2}+2K & -4K-4K-2 \\ -2{K}^{2}-2{K}^{2}-K & 2K+4{K}^{2}+1+4K\end{matrix}]$$=[4K2+2K−2K2−2K2−K​−4K−4K−22K+4K2+1+4K​]

$${A}^{4}+3\mathrm{AI}=2I$$A4+3AI=2I

$$2I=[\begin{matrix}4{K}^{2}+2K & -8K-2 \\ -4{K}^{2}-K & 4{K}^{2}+6K+1\end{matrix}]+3[\begin{matrix}0 & 2 \\ K & -1\end{matrix}][\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}]$$2I=[4K2+2K−4K2−K​−8K−24K2+6K+1​]+3[0K​2−1​][10​01​]

$$2I=[\begin{matrix}4{K}^{2}+2K & -8K-2 \\ -4{K}^{2}-K & 4{K}^{2}+6K+1\end{matrix}]+3[\begin{matrix}0 & 2 \\ K & -1\end{matrix}]$$2I=[4K2+2K−4K2−K​−8K−24K2+6K+1​]+3[0K​2−1​]

$$2I=[\begin{matrix}4{K}^{2}+2K & 6-8K-2 \\ 3K-K-4{K}^{2} & 4{K}^{2}+6K+1-3\end{matrix}]$$2I=[4K2+2K3K−K−4K2​6−8K−24K2+6K+1−3​]

$$4-8K=0$$4−8K=0

$$K=\frac{1}{2}$$K=21​