If A and B are square n × n matrices such that

$\mathbf{AB=I}$ | (16) |

then B is a right inverse of A. Similarly, if C is an n × n matrix such that

$\mathbf{CA=I}$ | (17) |

then C is a left inverse of A. When both Equation 16 and Equation 17 hold

$\mathbf{AB=CA=I}$ | (18) |

then B = C and B is the two-sided inverse of A.

The two-sided inverse of A will be referred to as its multiplicative
inverse or simply its inverse. If the inverse of A exists, it is
unique and denoted by A^{-1}. A^{-1} exists if and
only if A is square and nonsingular. A square n ×n
matrix is singular when its rank is less than n, i.e. two or more of its columns
(or rows) are linearly dependent. The rank of a matrix is examined more closely in
Section 2.7 of this document.

A few additional facts about inverses. If A is invertible, so is
A^{-1} and

$\mathbf{{\left({A}^{-1}\right)}^{-1}=A}$ | (19) |

If A and B are invertible, so is AB and

$\mathbf{{\left(AB\right)}^{-1}={B}^{-1}{A}^{-1}}$ | (20) |

Extending the previous example

$\mathbf{{\left(ABC\right)}^{-1}={C}^{-1}{B}^{-1}{A}^{-1}}$ | (21) |

If A is invertible, then

$\mathbf{{\left({A}^{-1}\right)}^{T}={\left({A}^{T}\right)}^{-1}}$ | (22) |

The conditional or generalized inverse which may be defined for any matrix is beyond the scope of the current discussion.