Inverse Matrix and Invertible Real Square Matrix

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $n\in\mathbb{N}$ and let $A\in\mathcal{M}_{n}(\mathbb{R})$ be a square real matrix, with the matrix product and the identity matrix $I_{n}$.

(i) A matrix $B\in\mathcal{M}_{n}(\mathbb{R})$ is an inverse of $A$ if $AB=I_{n}$ and $BA=I_{n}$.

(ii) $A$ is invertible if it has an inverse. An invertible $A$ has exactly one inverse: if $B$ and $C$ are inverses of $A$, then $AB=I_{n}$ and $CA=I_{n}$, so $B=C$ by Uniqueness of the Matrix Inverse (i). This inverse is denoted $A^{-1}$.