The Determinant function returns the mod m determinant of the input square mod m Matrix.

The following methods are available:

Note that the two inplace methods available will destroy the data in the input Matrix, while the other 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.

This command is part of the LinearAlgebra[Modular] package, so it can be used in the form Determinant(..) only after executing the command with(LinearAlgebra[Modular]).  However, it can always be used in the form LinearAlgebra[Modular][Determinant](..).

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

$p≔97$

$p≔97$

$M≔\mathrm{Create}\left(p\,4\,4\,\mathrm{random}\,\mathrm{float}\left[8\right]\right)$

$M≔\left[\begin{array}{cccc}96.& 62.& 93.& 90.\\ 58.& 16.& 76.& 19.\\ 7.& 49.& 42.& 78.\\ 26.& 19.& 21.& 21.\end{array}\right]$

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

$5,\left[\begin{array}{cccc}96.& 62.& 93.& 90.\\ 58.& 16.& 76.& 19.\\ 7.& 49.& 42.& 78.\\ 26.& 19.& 21.& 21.\end{array}\right]$

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

$5,\left[\begin{array}{cccc}1.& 35.& 4.& 7.\\ 0.& 1.& 86.& 38.\\ 0.& 0.& 1.& 96.\\ 0.& 0.& 0.& 1.\end{array}\right]$

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

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

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

$5$

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

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