The multilevel fast multipole algorithm (MLFMA) is traditionally employed in the context of an iterative matrix solver, in which the MLFMA is utilized to implement the underlying matrix product with N log N complexity, where N represents the number of unknowns. The total computational complexity of such an approach is order P N log N, where P represents the number of iterations required for iterative-solver (e.g. conjugate gradients) convergence to a desired accuracy. Many electromagnetic-scattering problems are poorly conditioned, and therefore P is often large. In this paper, rather than applying an iterative matrix solver, we perform a matrix product involving the inverse of the impedance matrix. By using the properties of the MLFMA, this process is performed very efficiently for electrically large problems. In particular, numerical experiments indicate that this new formulation (which avoids the iteration count P) is often significantly faster than the traditional iterative MLFMA solution, while requiring the same computer memory. The basic theory is described, and several examples are presented.