# 2.6 Inverse of a Matrix

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.