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)|
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.