In order to investigate the form of the van der Waals interaction in different multilayer geometries we reformulate the Lifshitz theory in terms of an algebra of 2x2 matrices. This device allows us to derive a closed form solution for the secular determinant of the modes in terms of simple quadratures with explicit N dependence. We specifically investigate (i) the van der Waals interactions between a substrate and a multilayer system as a function of the separation between the substrate and the multilayer system and (ii) the interaction between two multilayer systems over a medium of variable separation. (C) 2003 American Institute of Physics.