  A lonesum matrix is a $(0,1)$-matrix that is uniquely determined by its row
and column sum vectors. In this paper, we introduce lonesum decomposable
matrices and study their properties. We provide a necessary and sufficient
condition for a matrix $A$ to be lonesum decomposable, and give a generating
function for the number $D_k(m,n)$ of $m\times n$ lonesum decomposable matrices
of order $k$. Moreover, by using this generating function we prove some
congruences for $D_k(m,n)$ modulo a prime.
