Question

Matrices $A$ and $B$ satisfy $AB=B^{-1}$, where $B=\left[\begin{array}{cc}2& -1\\ 2& 0\end{array}\right].$ Then the value of the scalar $k$ for which $kA-2B^{-1}+I=O$ is

Answer 1

Given: $AB=B^{-1}$
$\Rightarrow A{B}^2=I\cdots \left(i\right)$

Now, $kA-2B^{-1}+I=O$
$\Rightarrow kAB-2I+B=O$
$\Rightarrow kA{B}^2-2B+B^2=O$
$\Rightarrow kI-2B+B^2=O\left(\text{Using}\right(i\left)\right)$
$\Rightarrow k\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]-2\left[\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]+\left[\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]\left[\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]=O$
$\Rightarrow \left[\begin{array}{cc}k& 0\\ 0& k\end{array}\right]-\left[\begin{array}{cc}4& -2\\ 4& 0\end{array}\right]+\left[\begin{array}{cc}2& -2\\ 4& -2\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}k-2& 0\\ 0& k-2\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$\Rightarrow k-2=0\Rightarrow k=2$