Question

Matrices A and B satisfy $AB=B^{-1}$, where $B=\left[\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]$, then find the value of $k$ for which $kA-2B^{-1}+I=O$, without finding $B^{-1}$

Answer 1

$MatricesAandBsatisfyAB=B^{-1},whereB=\left[\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]kA-2B^{-1}+I=O$

$\Rightarrow kAB-2B^{-1}B+IB=O$

$\Rightarrow k{B}^{-1}-2I+B=O$

$\Rightarrow k{B}^{-1}B-2IB+BB=O$

$kI-2B+B^2=O$

$kI=2B-B^2$

$\left[\begin{array}{cc}k& 0\\ 0& k\end{array}\right]=2\left[\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]-\left[\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]\left[\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]$

$comparingelementsonbothsides$

$k=4-2=2$

$Hence,k=2$