Matrices A and B satisfy $AB = B^{-1}$, where $B =\begin{bmatrix}2&-1\\2&0\end{bmatrix}$. Find the value of K for which $KA-2B^{-1}+I=O$.

2

$AB = B^{-1}$ or $ABB = B^{-1}B$ or $AB^2 = I$

Now, $KA-2B^{-1}+I=O$

or $KAB-2B^{-1}B+IB = O$

or $KAB-2I+B=O$

or $KAB^2-2B+ B^2 =O$

or $KI-2B+B^2=O$

or $K\begin{bmatrix}1&0\\0&1\end{bmatrix}-2\begin{bmatrix}2&-1\\2&0\end{bmatrix}+\begin{bmatrix}2&-1\\2&0\end{bmatrix}\begin{bmatrix}2&-1\\2&0\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}$

or $\begin{bmatrix}K&0\\0&K\end{bmatrix}-\begin{bmatrix}4&-2\\4&0\end{bmatrix}+\begin{bmatrix}2&-2\\4&-2\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}$

or $\begin{bmatrix}K-2&0\\0&K-2\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}$

or $K=2$