Definition:Invertible Matrix

Definition
Let $$\left({R, +, \circ}\right)$$ be a ring with unity.

Let $$\mathcal M_R \left({n}\right)$$ be the $n \times n$ matrix space over $$R$$.

Let $$\mathbf A$$ be an element of the ring $\left({\mathcal M_R \left({n}\right), +, \times}\right)$.

Then $$\mathbf A$$ is invertible iff:
 * $$\exists \mathbf B \in \left({\mathcal M_R \left({n}\right), +, \times}\right): \mathbf A \mathbf B = \mathbf{I_n} = \mathbf B \mathbf A$$

Such a $$\mathbf B$$ is the inverse of $$\mathbf A$$. It is usually denoted $$\mathbf A^{-1}$$.

If a matrix has no such inverse, it is called non-invertible.

As $$\left({R, +, \circ}\right)$$ is a ring with unity, it follows from Inverses are Unique that the inverse of a matrix is unique.

It also follows from Inverse of Product that if both $$\mathbf A$$ and $$\mathbf B$$ are invertible matrices, then so is $$\mathbf A \mathbf B$$, and its inverse is $$\mathbf B^{-1} \mathbf A^{-1}$$.

Comment
Some authors use the term singular to mean non-invertible, and likewise non-singular to mean invertible.

The term regular is also sometimes found for invertible.