Proof of Uniqueness of the Matrix Inverse

Suppose $B$ and $C$ are both inverses of $A$, so

$$AB = BA = I_n \quad\text{and}\quad AC = CA = I_n,$$

where $I_n$ is the identity matrix and the products are taken in the sense of the matrix product definition. By associativity of the matrix product,

$$B = B I_n = B(AC) = (BA)C = I_n C = C.$$

Hence $B = C$, and the inverse is unique.