If A and B are square n × n matrices such that
then B is a right inverse of A. Similarly, if C is an n × n matrix such that
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
If A and B are invertible, so is AB and
Extending the previous example
If A is invertible, then
The conditional or generalized inverse which may be defined for any matrix is beyond the scope of the current discussion.