The purpose of this article is to present a general, computationally efficient, rank r matrix modification scheme for the solution of the linear matrix equation \(H'x'=u'\); it is assumed that the matrix H' differs by a matrix of low rank from a matrix H of a system whose solution is known, or easily computed. The approach is based on a sum decomposition of H'-H in terms of rank 1 matrices. The method explicitly manifests the role of physical parameters characterizing the system, and finds several applications, for example to the numerical solution of partial differential equations, to the interpolation of cubic splines, or in the control systems area, applications briefly discussed in the paper. For the low rank case \(r=2\), the modification equations are solved analytically, and the operational count is compared with that of previous algorithms. Finally, an analytical form is developed for the inverse of a nonsymmetric tridiagonal Toeplitz matrix with arbitrary corner elements.