The Rank function returns the rank of the input mod m Matrix, while the RankProfile function returns a list of 'rank' elements describing the rank profile of the input mod m Matrix.

The rank profile list is simply a list of the location of the first non-zero entry in each nontrivial row in the row reduced form of the Matrix.

The following methods are available:

Note that the two inplace methods available will destroy the data in the input Matrix, while the other two methods will generate a copy of the Matrix in which to perform the computation.

The RET methods are likely to be faster for large matrices, but may fail if the modulus is composite.

These commands are part of the LinearAlgebra[Modular] package, so they can be used in the form Rank(..) and RankProfile(..) only after executing the command with(LinearAlgebra[Modular]).  However, they can always be used in the form LinearAlgebra[Modular][Rank](..) and LinearAlgebra[Modular][RankProfile](..).

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

$p≔97$

$p≔97$

$M≔\mathrm{Mod}\left(p\,\left[\left[2\,11\,1\,80\right]\,\left[31\,16\,27\,36\right]\,\left[8\,32\,32\,31\right]\,\left[25\,4\,90\,63\right]\right]\,{\mathrm{integer}}_{\[\]}\right)$

$M≔\left[\begin{array}{cccc}2& 11& 1& 80\\ 31& 16& 27& 36\\ 8& 32& 32& 31\\ 25& 4& 90& 63\end{array}\right]$

$\mathrm{Rank}\left(p\,M\right)\,\mathrm{RankProfile}\left(p\,M\right)$

$4,\left[1\,2\,3\,4\right]$

$\mathrm{Rank}\left(p\,M\,\mathrm{inplaceREF}\right)\,M$

$4,\left[\begin{array}{cccc}1& 54& 49& 40\\ 0& 1& 58& 26\\ 0& 0& 1& 35\\ 0& 0& 0& 1\end{array}\right]$

$M≔\mathrm{Mod}\left(p\,\left[\left[2\,2\,1\,80\right]\,\left[31\,31\,27\,36\right]\,\left[8\,8\,32\,31\right]\,\left[25\,25\,90\,63\right]\right]\,{\mathrm{integer}}_{\[\]}\right)$

$M≔\left[\begin{array}{cccc}2& 2& 1& 80\\ 31& 31& 27& 36\\ 8& 8& 32& 31\\ 25& 25& 90& 63\end{array}\right]$

$\mathrm{RankProfile}\left(p\,M\right)$

$\left[1\,3\,4\right]$

$\mathrm{Rank}\left(p\,M\right)$

$3$

$M≔\mathrm{Matrix}\left(\left[\left[3\,2\right]\,\left[2\,3\right]\right]\,\mathrm{datatype}\=\mathrm{integer}\right)$

$M≔\left[\begin{array}{cc}3& 2\\ 2& 3\end{array}\right]$

$\mathrm{Rank}\left(6\,M\,\mathrm{REF}\right)$

$2$

$\mathrm{Rank}\left(6\,M\,\mathrm{RET}\right)$

Error, (in LinearAlgebra:-Modular:-RowEchelonTransform) modulus is composite