To describe the dynamics of stage-structured populations with m stages living in n patches, we consider matrix models of the form SD where S is a block diagonal matrix with n × n column substochastic matrices S1, … , Sm along the diagonal and D is a block

Generalizing a well known trace formula from linear algebra, we define a generalized determinant of a pair of endomorphisms in arbitrary (infinite) dimensional vector spaces. We prove, in a purely linear algebraic context, that this generalized determinant is a true determinant and that any formal power series with rational coefficients can be seen as one of these determinants. This result was already given a different proof in the Ph.D. thesis of the first author, but was not available in the literature, yet. Some illustrations are given regarding the study of the dynamical zeta function of an interval map.