The Gausselim and Gaussjord functions are placeholders for representing row echelon forms of the rectangular matrix A.

The commands Gausselim(A,...) mod p and Gassjord(A,...) mod p apply Gaussian elimination with row pivoting to A, a rectangular matrix over a finite ring of characteristic p. This includes finite fields, GF(p), the integers mod p, and GF(p^k) where elements of GF(p^k) are expressed as polynomials in RootOfs.

The result of the Gausselim command is a an upper triangular matrix B in row echelon form.  The result of the Gaussjord command is also an upper triangular matrix B but in reduced row echelon form.

If an optional second parameter is specified, and it is a name, it is assigned the rank of the matrix A.

If A is an $m$ by $n$ matrix with $m\le n$ and if an optional third parameter is also specified, and it is a name, it is assigned the determinant of the matrix A[1..m,1..m].

$A≔\mathrm{Matrix}\left(\left[\left[1\,2\,3\right]\,\left[1\,3\,0\right]\,\left[1\,4\,3\right]\right]\right)$

$A≔\left[\begin{array}{ccc}1& 2& 3\\ 1& 3& 0\\ 1& 4& 3\end{array}\right]$

$\mathrm{Gausselim}\left(A\right)\mathbf{mod}5$

$\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 2\\ 0& 0& 1\end{array}\right]$

$B≔\mathrm{ArrayTools}\left[\mathrm{Concatenate}\right]\left(2\,A\,\mathrm{LinearAlgebra}\left[\mathrm{IdentityMatrix}\right]\left(3\right)\right)$

$B≔\left[\begin{array}{cccccc}1& 2& 3& 1& 0& 0\\ 1& 3& 0& 0& 1& 0\\ 1& 4& 3& 0& 0& 1\end{array}\right]$

$\mathrm{Gaussjord}\left(B\right)\mathbf{mod}5$

$\left[\begin{array}{cccccc}1& 0& 0& 4& 1& 1\\ 0& 1& 0& 2& 0& 3\\ 0& 0& 1& 1& 3& 1\end{array}\right]$

$\mathrm{Inverse}\left(A\right)\mathbf{mod}5$

$\left[\begin{array}{ccc}4& 1& 1\\ 2& 0& 3\\ 1& 3& 1\end{array}\right]$

$\mathrm{alias}\left(a=\mathrm{RootOf}\left(x^4+x+1\right)\mathbf{mod}2\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[1\,a\,a^2\right]\,\left[a\,a^2\,a^3\right]\,\left[a^2\,a^3\,1\right]\right]\right)$

$A≔\left[\begin{array}{ccc}1& a& a^2\\ a& a^2& a^3\\ a^2& a^3& 1\end{array}\right]$

$\mathrm{Gausselim}\left(A\,'r'\,'d'\right)\mathbf{mod}2$

$\left[\begin{array}{ccc}1& a& a^2\\ 0& 0& a\\ 0& 0& 0\end{array}\right]$

$r$

$2$

$d$

$0$