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  
\[A^{-1}AR=A^{-1} I \rightarrow IR =A^{-1} \rightarrow R=A^{-1}\]
.