Lattice matrices are 0/1-matrices used in the description of certain lattice polyhedra and related to dicuts in a graph. The incidence matrix A^I of a so-called intersection of two ring families and the incidence matrix A^D of all dicuts of a graph are examples of such matrices. After showing that any lattice matrix A can be obtained from some matrix A^I by deletion or some A^D by contraction, we first describe the convex hull of the rows of A^I, CONV(A^I), as the solution set of a system x⩾0, Bx⩽1, Rx⩽0, which is tdi. We then derive the main result, the description of CONV(A) by another tdi system. As applications, the polyhedral description of all dicuts in a graph, CONV(A^D), and that of all convex sets of bounded length in a poset are established.