2.6 Inverse of a Matrix

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

𝐀𝐁=𝐈\mathbf{AB=I} (16)

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

𝐂𝐀=𝐈\mathbf{CA=I} (17)

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

𝐀𝐁=𝐂𝐀=𝐈\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

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

If A and B are invertible, so is AB and

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

Extending the previous example

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

If A is invertible, then

(𝐀-𝟏)𝐓=(𝐀𝐓)-𝟏\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.