The propagator matrix formalism of F. Gilbert and G. E. Backus is a convenient general method for treating stratified media. In this paper, the Gilbert–Backus formalism is used in conjunction with Sylvester’s matrix theorem to obtain an exact solution for wave propagation in a periodically stratified medium of arbitrary thickness. Explicit expressions are derived for the simple case of a periodically layered fluid. These expressions are used to investigate several interesting limiting cases. Application of the method to the more complicated case of a periodically layered solid is briefly discussed.