The Pivot(A, i, j) command pivots A about the nonzero entry $A_{i,j}$.  Multiples of the ith row are added to every other row in A, with the result that all entries in the jth column of A are zero except for the $i,j$th element.

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

$A≔⟨⟨1\,5\,9\,3⟩\left|⟨2\,6\,0\,4⟩\right|⟨3\,7\,1\,5⟩|⟨4\,8\,2\,6⟩⟩$

$A≔\left[\begin{array}{cccc}1& 2& 3& 4\\ 5& 6& 7& 8\\ 9& 0& 1& 2\\ 3& 4& 5& 6\end{array}\right]$

$P≔\mathrm{Pivot}\left(A\,1\,1\right)$

$P≔\left[\begin{array}{cccc}1& 2& 3& 4\\ 0& \mathrm{-4}& \mathrm{-8}& \mathrm{-12}\\ 0& \mathrm{-18}& \mathrm{-26}& \mathrm{-34}\\ 0& \mathrm{-2}& \mathrm{-4}& \mathrm{-6}\end{array}\right]$

$\mathrm{Pivot}\left(P\,2\,2\right)$

$\left[\begin{array}{cccc}1& 0& \mathrm{-1}& \mathrm{-2}\\ 0& \mathrm{-4}& \mathrm{-8}& \mathrm{-12}\\ 0& 0& 10& 20\\ 0& 0& 0& 0\end{array}\right]$