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In sage, the syntax P, L, U = A.LU(pivot='nonzero') gives the
matrix equation A=PLU where P is a permutation
matrix, L is lower-triangular, and U is
upper-triangular. We can redefine P as its
transpose P=P.T to obtain PA=LU.
In class, we require U be in row echelon form, not just upper
triangular. So, we may need to do a few more reductions ourselves.