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one of and has to be zero, which implies that either or is singular. This is impossible if is nonsingular (invertible). In terms of operations, zeroing/elimination of remaining elements of first column of involves division of with , impossible if it is 0. This is a procedural problem. It can be removed by simply reordering the rows of so that the first element of the permuted matrix is nonzero. The same problem in subsequent factorization steps can be removed the same way. For numerical stability

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