# Definition:Invertible Matrix

## Definition

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 $\struct {\map {\mathcal M_R} n, +, \times}$.

Then $\mathbf A$ is **invertible** if and only if:

- $\exists \mathbf B \in \struct {\map {\mathcal M_R} n, +, \times}: \mathbf A \mathbf B = \mathbf I_n = \mathbf B \mathbf A$

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 $\mathbf A^{-1}$.

As $\struct {R, +, \circ}$ is a ring with unity, it follows from Product Inverse in Ring is Unique that the inverse of a matrix is unique.

## Non-Invertible Matrix

Let $\mathbf A$ have no inverse.

Then $\mathbf A$ is referred to as **non-invertible**.

## Also known as

An **invertible matrix** is called by some authors a **non-singular matrix** or **regular matrix**.

## Also see

- Inverse of Matrix Product: if both $\mathbf A$ and $\mathbf B$ are invertible matrices, then so is $\mathbf A \mathbf B$, and its inverse is $\mathbf B^{-1} \mathbf A^{-1}$.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 29$ - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.2$: Functions on vectors: $\S 2.2.5$: Determinants