The IntegerDeterminant function computes the determinant of the integer matrix M. This is a programmer level function, it does not perform argument checking. Thus, argument checking must be handled external to this function.

Note: The IntegerDeterminant command uses a probabilistic approach that achieves great gains for structured systems. Information on controlling the probabilistic behavior can be found in _EnvProbabilistic.

This function is used by the Determinant function in the LinearAlgebra package when a Matrix is determined to contain only integer entries.

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

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

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

$M≔\left[\begin{array}{ccc}2& 1& 3\\ 4& 3& 1\\ \mathrm{-2}& 1& \mathrm{-3}\end{array}\right]$

$\mathrm{IntegerDeterminant}\left(M\right)$

$20$

$M≔\mathrm{LinearAlgebra}\left[\mathrm{RandomMatrix}\right]\left(100\right)\:$

$\mathrm{tt}≔\mathrm{time}\left(\right)\:$

$\mathrm{IntegerDeterminant}\left(M\right)$

$38562295347802366242417909657285032281105091485000162871067163275296273582728190925949289361981964881806516849833008824879568403928373759144147382030798909099402726531205056808283212790472544339698767179236612577117605985054960334148934541347201762137455$

$\mathrm{time}\left(\right)-\mathrm{tt}$

$0.028$