Question

Matrices A and B satisfy $AB=B^{-1}$, where $B=\left[\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]$, then find without finding $A^{-1}$, the matrix X satisfying $A^{-1}XA=?$

• A. $B$
• B. $B^2$
• C. $A$
• D. None of these

Answer 1

The correct option is A $B$

Given, $A^{-1}XA=B$

$A{A}^{-1}XA=AB$

$IXA=AB$

$XAB=A{B}^2$

$XAB=I$ since $[AB=B^{-1}\Rightarrow A{B}^2=I]$

$XA{B}^2=B$

$XI=B$

$\therefore X=B$

Hence option ${}^{\prime }{A}^{\prime }$ is the answer.