Properties of Inverse Matrix

Theorem

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

Proof

Let $B,C$ be inverse of $A$. Then we have $BA=I$.

Multiplying both sizes by $C$ gives $(BA)C=IC=C$.

But it is also true that $(BA)C=B(AC)=BI=B$.

Thus, $B=C$.

Remark

As a consequence of this theorem, if $A$ is invertible, we simply denote inverse of $A$ as $A^{-1}$. Thus,

$$A{A}^{-1}=A^{-1}A=I$$