# 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 $\mathbf B^{-1} \mathbf A^{-1}$.