Proof of Uniqueness of the Matrix Inverse

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

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

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.