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 Product Inverses in Ring 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.