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} = (A^{T} ) ^{–1}

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