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 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.