The problem of transforming a given sparse matrix A into a block diagonal form (b.d.f.) and the subsequent transformation of each of the block diagonal matrices into as nearly upper triangular form (u.t.f.) as possible, by using only row and column permutations, is discussed. It is shown how some of the results from Graph Theory can be used to transform A to the b.d.f. In order to transform the block diagonal matrices into the u.t.f.'s, two methods are described, one of which makes use of linear programming while the other uses approximate probabilistic arguments. The latter method, in relation to the computational effort, yields significant results in practice.