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Some authors use the term Gaussian elimination to refer to the process until it has reached its upper triangular, or (non-reduced) row echelon form. For computational reasons, when solving systems of linear equations, it is sometimes preferable to stop row operations before the matrix is completely reduced.
As explained above, Gaussian elimination writes a given Ã matrix uniquely as a product of an invertible Ã matrix and a row-echelon matrix. Here, is the product of the matrices corresponding to the row operations performed. The formal algorithm to compute from follows.