The GaussianElimination(A) command performs Gaussian elimination on the Matrix A and returns the upper triangular factor U with the same dimensions as A.

This command is equivalent to calling LUDecomposition with the output=['U'] option.

The ReducedRowEchelonForm(A) command performs Gauss-Jordan elimination on the Matrix A and returns the unique reduced row echelon form R of A.

This command is equivalent to calling LUDecomposition with the output=['R'] option.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$A≔⟨⟨8\,3\,-1\,-5⟩\left|⟨4\,-5\,0\,-2⟩\right|⟨-5\,8\,3\,-1⟩|⟨-5\,5\,-4\,-9⟩⟩$

$A≔\left[\begin{array}{cccc}8& 4& \mathrm{-5}& \mathrm{-5}\\ 3& \mathrm{-5}& 8& 5\\ \mathrm{-1}& 0& 3& \mathrm{-4}\\ \mathrm{-5}& \mathrm{-2}& \mathrm{-1}& \mathrm{-9}\end{array}\right]$

$b≔⟨4\,0\,-8\,-5⟩$

$b≔\left[\begin{array}{c}4\\ 0\\ \mathrm{-8}\\ \mathrm{-5}\end{array}\right]$

$\mathrm{GaussianElimination}\left(A\right)$

$\left[\begin{array}{cccc}8& 4& \mathrm{-5}& \mathrm{-5}\\ 0& -\frac{13}{2}& \frac{79}{8}& \frac{55}{8}\\ 0& 0& \frac{163}{52}& -\frac{213}{52}\\ 0& 0& 0& -\frac{2607}{163}\end{array}\right]$

$\mathrm{ReducedRowEchelonForm}\left(⟨A|b⟩\right)$

$\left[\begin{array}{ccccc}1& 0& 0& 0& \frac{1715}{2607}\\ 0& 1& 0& 0& -\frac{3668}{2607}\\ 0& 0& 1& 0& -\frac{1345}{869}\\ 0& 0& 0& 1& \frac{1759}{2607}\end{array}\right]$