Three mysteries of Gaussian elimination

If numerical analysts understand anything, surely it must be Gaussian elimination. This is the oldest and truest of numerical algorithms. To be precise, I am speaking of Gaussian elimination with partial pivoting, the universal method for solving a dense, unstructured n X n linear system of equations Ax = b on a serial computer. This algorithm has been so successful that to many of us, Gaussian elimination and Ax = b are more or less synonymous. The chapter headings in the book by Golub and Van Loan [3] are typical -- along with "Orthogonalization and Least Squares Methods," "The Symetric Eigenvalue Problem," and the rest, one finds "Gaussian Elimination," not "Linear Systems of Equations."