A matrix problem

Problem

\(A,B,C\) are three square invertible matrices. Show that

\[{\left( AB^{-1}C \right)}^{-1}=C^{-1}BA^{-1}\]

Hint: Remember that if \(AB=I\) then \(B\) must be the inverse of \(A\) . So you need to show that

\[\left( AB^{-1}C \right)⋅C^{-1}BA^{-1}=I\]

It then follows that \(C^{-1}BA^{-1}\) is the inverse of \(AB^{-1}C\) .

Solution

\[AB^{-1}CC^{-1}BA^{-1}=AB^{-1}BA^{-1}\]

since \(CC^{-1}=I\) and

\[AB^{-1}BA^{-1}=AA^{-1}=I\]

since \(B^{-1}B=I\) and \(AA^{-1}=I\) .