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. Then

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

where the first and last equalities hold because the identity matrix is a two-sided multiplicative identity, by The Identity Matrix is a Two-Sided Multiplicative Identity; the second and fourth equalities substitute $AC=I_n$ and $BA=I_n$ from the inverse property; and the third equality is associativity of the matrix product. Hence $B = C$, and the inverse is unique.