That the inverse matrix of $A$ is unique means that there is only one inverse matrix of $A$.
(That’s why we say “the” inverse matrix of $A$ and denote it by $A^{-1}$.)
So to prove the uniqueness, suppose that you have two inverse matrices $B$ and $C$ and show that in fact $B=C$.

Recall that $B$ is the inverse matrix if it satisfies
\[AB=BA=I,\]
where $I$ is the identity matrix.

Proof.

Suppose that there are two inverse matrices $B$ and $C$ of the matrix $A$. Then they satisfy
\[AB=BA=I \tag{*}\]
and
\[AC=CA=I \tag{**}.\]

To show that the uniqueness of the inverse matrix, we show that $B=C$ as follows. Let $I$ be the $n\times n$ identity matrix.
We have
\begin{align*}
B&=BI\\
&=B(AC) && \text{ by (**)}\\
&=(BA)C &&\text{ by the associativity}\\
&=IC && \text{ by (*)}\\
&=C.
\end{align*}
Thus, we must have $B=C$, and there is only one inverse matrix of $A$.

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