Matrix Inverses are unique

Left Inverse = Right Inverse

Let \( A \) be an \( n \times n \) matrix and \( AB = \mathbb{I} \), so that \( B = A^{-1} \).
It follows that \( BA = \mathbb{I} \)...
\begin{aligned} AB & = \mathbb{I} \\ B \left( AB \right) & = B \mathbb{I} \\ \left( B A \right) B & = B \\ \therefore \left( B A \right) & = \mathbb{I} \end{aligned}

Inverses Are Unique

Suppose another inverse of \( A \) existed, \( C \), so that \( AC = CA = \mathbb{I} \).
It follows that \( B = C = A^{-1} \)...
\begin{aligned} A B & = \mathbb{I} \\ C \left( A B \right) & = C \mathbb{I} \\ \left( C A \right) B & = C \\ \left( \mathbb{I} \right) B & = C \\ \therefore B & = C \end{aligned}