Definition:Nonsingular 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 $\struct {\map {\MM_R} n, +, \times}$.

Definition 1

$\mathbf A$ is a nonsingular matrix if and only if:

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

where $\mathbf I_n$ denotes the unit matrix of order $n$.

Definition 2

$\mathbf A$ is a nonsingular matrix if and only if:

$\det {\mathbf A} \ne 0$

where $\det$ denotes the determinant.

Definition 3

$\mathbf A$ is a nonsingular matrix if and only if $\mathbf A$ is not singular.

Also known as

Nonsingular matrix can also be seen hyphenated: non-singular matrix.

A nonsingular matrix is also called by some authors:

an invertible matrix

a regular matrix.

Also see

• Definition:Inverse Matrix

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

• Inverse of Square Matrix over Field is Unique: the inverse of a nonsingular matrix is unique.

• Definition:Singular Matrix, also known as a non-invertible matrix

• Results about nonsingular matrices can be found here.