By a nearly diagonal matrix is meant a matrix in which the off-diagonal elements are small compared with those on the diagonal. Such matrices occur, for instance, in connection with non-orthogonal experimental designs. Inversion may be performed by methods of general applicability such as the Doolittle or Cholesky procedures (direct methods) or relaxation (an iterative method); many such procedures are described in the literature. Those already familiar with any of these methods may well prefer them to the one described below because of their wider applicability, but anyone not so placed who is often concerned with the inversion of nearly diagonal matrices should find the method described here useful.