Definition:Invertible Matrix

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" for "non-invertible", and likewise "non-singular" for "invertible".

The term "regular" is also sometimes found for "invertible".