\begin{definition}[Definition:Invertible Matrix]
Let $\struct {R, +, \circ}$ be a ring with unity.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\mathbf A$ be an element of the ring of square matrices $\struct {\map {\MM_R} n, +, \times}$.
Then $\mathbf A$ is '''invertible''' {{iff}}:
:$\exists \mathbf B \in \struct {\map {\MM_R} n, +, \times}: \mathbf A \mathbf B = \mathbf I_n = \mathbf B \mathbf A$
where $\mathbf I_n$ denotes the unit matrix of order $n$.
Such a $\mathbf B$ is the inverse of $\mathbf A$.
It is usually denoted $\mathbf A^{-1}$.
\end{definition}