Proof that inverse of a matrix is unique

Question

If B and C are both inverses of the matrix A,then B=C.

Can't i prove it in following way ?

Proof:

AB=BA=I and AC=CA=I,then BA=CA=I

By postmultiplication $\Rightarrow (BA)(A^{-1})=(CA)(A^{-1})=(I)(A^{-1})\Rightarrow B=C=A^{-1}$,

or by premultiplication $AB=AC=I\Rightarrow (A^{-1})(AB)=(A^{-1})(AC)=(A^{-1})(I)\Rightarrow B=C=A^{-1}$.

Answer 1

There is much much simpler.

$B=BI=B(AC)=(BA)C=IC=C$