  In this paper we develop a method to compute the solution to a countable
(finite or infinite) set of equations that occurs in many different fields
including Markov processes that model queueing systems, birth-and-death
processes and inventory systems. The method provides a fast and exact
computation of the inverse of the matrix of the coefficients of the system. In
contrast, alternative inverse techniques perform much slower and work only for
finite size matrices. Furthermore, we provide a procedure to construct the
eigenvalues of the matrix under consideration.
