## Proof That the Left Inverse of a Matrix Equals the Right Inverse

Theorem
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
$A^{-1}AR=A^{-1} I \rightarrow IR =A^{-1} \rightarrow R=A^{-1}$
.