At the end of Chapter 4 we discovered matrices E and F with the property that EF = I and FE = I, and we said that they were inverses of each other. Generally, if A is a square matrix and B is a matrix of the same size with AB = I and BA = I, then B is said to be the inverse of A. The inverse of A is denoted by A–1. Example 5.1 is a proof that the inverse of a matrix (if it exists at all) is unique. Example 5.2 gives a matrix which does not have an inverse. So we must take care: not every matrix has an inverse. A matrix which does have an inverse is said to be invertible (or non-singular). Note that an invertible matrix must be square. A square matrix which is not invertible is said to be singular.

Following our discussion in Chapter 4 we can say that every elementary matrix is invertible and every orthogonal matrix is invertible. Example 5.3 shows that every diagonal matrix with no zeros on the main diagonal is invertible. There is, however, a standard procedure for testing whether a given matrix is invertible, and, if it is, of finding its inverse. This process is described in this chapter. It is an extension of the GE process.