  A matrix is called acyclic if replacing the diagonal entries with $0$, and
the nonzero diagonal entries with $1$, yields the adjacency matrix of a forest.
In this paper we show that null space and the rank of a acyclic matrix with $0$
in the diagonal is obtained from the null space and the rank of the adjacency
matrix of the forest by multipliying by non-singular diagonal matrices. We
combine these methods with an algorithm for finding a sparsest basis of the
null space of a forest to provide an optimal time algorithm for finding a
sparsest basis of the null space of acyclic matrices with $0$ in the diagonal.
