Question

If $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right];B=\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]$, then show that ${(A.B)}^{-1}=B^{-1}.A^{-1}$

Answer 1

Since $A$ is the identity matrix, $A=A^{-1}=I\Rightarrow AB=B$.

$\Rightarrow (AB{)}^{-1}=B^{-1}$

$\Rightarrow B^{-1}{A}^{-1}=B^{-1}$

$\Rightarrow (AB{)}^{-1}=B^{-1}{A}^{-1}$

Hence proved.