# Definition:Inverse Matrix

## Definition

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\mathbf A$ be a square matrix of order $n$.

Let there exist a square matrix $\mathbf B$ of order $n$ such that:

$\mathbf A \mathbf B = \mathbf I_n = \mathbf B \mathbf A$

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

Then $\mathbf B$ is called the inverse of $\mathbf A$ and is usually denoted $\mathbf A^{-1}$.

### Left Inverse Matrix

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix of order $m \times n$.

Let $\mathbf B = \sqbrk b_{n m}$ be a matrix of order $n \times m$ such that:

$\mathbf B \mathbf A = I_n$

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

Then $\mathbf B$ is known as a left inverse (matrix) of $\mathbf A$.

### Right Inverse Matrix

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix of order $m \times n$.

Let $\mathbf B = \sqbrk b_{n m}$ be a matrix of order $n \times m$ such that:

$\mathbf A \mathbf B = I_m$

where $I_m$ denotes the unit matrix of order $m$.

Then $\mathbf B$ is known as a right inverse (matrix) of $\mathbf A$.

## Also known as

An inverse matrix can also be seen referred to as a reciprocal matrix.

## Also see

• Results about inverse matrices can be found here.