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](..).

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

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

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

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

$20$

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

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

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

$38562295347802366242417909657285032281105091485000162871067163275296273582728190925949289361981964881806516849833008824879568403928373759144147382030798909099402726531205056808283212790472544339698767179236612577117605985054960334148934541347201762137455$

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

$0.042$