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