Let $A,B\in {\mathbb{R}}^{n\times n}$ be square, nonsingular matrices. Then, the following properties apply:

1. The matrix $A^{-1}$ is unique.
2. The linear-system problem $A\overrightarrow{x}=\overrightarrow{b}$ has a unique solution.
3. $A\cdot B$ is invertible and $(A\cdot B{)}^{-1}=B^{-1}\cdot A^{-1}$
4. $(A^T{)}^{-1}=(A^{-1}{)}^T=A^{-T}$
5. $A^{-1}$ can be written as a product of elementary matrices.