Rules for Matrix Inverses

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 $A,B,C\in\mathcal{M}_{n}(\mathbb{R})$, square real matrices, with the matrix product, the matrix-vector product, the transpose, the identity matrix $I_{n}$, invertible matrices and their inverses, and points of $\mathbb{R}^{n}$.

¶ $I_{n}$ is invertible and $I_{n}^{-1}=I_{n}$.

¶ If $AC=I_{n}$ or $CA=I_{n}$, then $A$ is invertible and $A^{-1}=C$.

¶ If $A$ is invertible, then $A^{-1}$ is invertible and $(A^{-1})^{-1}=A$.

¶ If $A$ and $B$ are invertible, then $AB$ is invertible and $(AB)^{-1}=B^{-1}A^{-1}$.

¶ If $A$ is invertible, then $A^{\top}$ is invertible and $(A^{\top})^{-1}=(A^{-1})^{\top}$.

¶ If $A$ is invertible and $b\in\mathbb{R}^{n}$, then $A^{-1}b$ is the only $x\in\mathbb{R}^{n}$ with $Ax=b$.