2.6 Inverse of a Matrix

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

AB = I (5)

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

CA = I (6)

then C is a left inverse of A. When both Equation 5 and Equation 6 hold

AB = CA = I

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

If A and B are invertible, so is AB and

(AB) –1 =  B–1 A–1

Extending the previous example

(ABC) –1 =  C–1 B–1 A–1

If A is invertible, then

(A–1 ) T =  (AT ) –1

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