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 function is equivalent to calling LinearAlgebra[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 function is equivalent to calling LinearAlgebra[LUDecomposition] with the output=['R'] option.

The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix constructor that builds the result. These options may also be provided in the form outputoptions=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions option, the latter takes precedence (regardless of the order).

These functions are part of the LinearAlgebra package, and so it can be used in the form GaussianElimination(..) or ReducedRowEchelonForm(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[GaussianElimination](..) or LinearAlgebra[ReducedRowEchelonForm](..).

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

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

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

$b≔⟨4\,0\,-8\,-5⟩$

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

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

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

$\mathrm{GaussianElimination}\left(A\,'\mathrm{method}'='\mathrm{FractionFree}'\right)$

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

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

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