If a square matrix
\[A\]
has a left inverse \[L\]
so that \[LA=I\]
and a right inverse \[R\]
so that \[AR=I\]
where \[I\]
is the identity matrix then \[L=R\]
.Proof
Let
\[L\]
be the left inverse of \[A\]
so that \[LA=I\]
.Composing on the right with
\[A^{-1}\]
gives \[LAA^{-1}=IA^{-1} \rightarrow LI =A^{-1} \rightarrow L=A^{-1}\]
.Let
\[R\]
be the right inverse of \[A\]
so that \[AR=I\]
.Composing on the left with
\[A^{-1}\]
gives \[A^{-1}AR=A^{-1} I \rightarrow IR =A^{-1} \rightarrow R=A^{-1}\]
.