Without a proper ordering or permutations in the matrix, the factorization may fail to materialize. For example, it is easy to verify (by expanding the matrix multiplication) that formula_3. If formula_4, then at least one of formula_5 and formula_6 has to be zero, which implies that either "L" or "U" is singular. This is impossible if "A" is nonsingular (invertible). This is a procedural problem. It can be removed by simply reordering the rows of "A" so that the first element of the permuted matrix is nonzero. The same problem in subsequent factorization steps can be removed the same way; see the basic procedure below.