The (indexed) parameter D specifies the domain of computation, an integral domain (a commutative ring with exact division). It must be a Maple table/module which has the following values/exports:

D[`0`] : a constant for the zero of the ring D

D[`1`] : a constant for the (multiplicative) identity of D

D[`+`] : a procedure for adding elements of D (nary)

D[`-`] : a procedure for negating and subtracting elements of D (unary and binary)

D[`*`] : a procedure for multiplying elements of D (binary and commutative)

D[`=`] : a boolean procedure for testing if two elements of D are equal

D[Divide] : a boolean procedure for testing if a | b in D, and if so assigns q the value of a / b.

BareissAlgorithm[D](A) runs Bareiss' fraction-free row reduction on a copy of A.

The output Matrix B is upper triangular, and the entry B[i,i] is the determinant of the principal i x i submatrix of A. Thus if A is a square Matrix of dimension n, then B[n,n] is the determinant of A up to a unit.

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

$Z\left[\mathrm{`0`}\right],Z\left[\mathrm{`1`}\right],Z\left[\mathrm{`+`}\right],Z\left[\mathrm{`-`}\right],Z\left[\mathrm{`*`}\right],Z\left[\mathrm{`=`}\right]≔0,1,\mathrm{`+`},\mathrm{`-`},\mathrm{`*`},\mathrm{`=`}\:$

Z[Divide] := proc(a,b,q) evalb( irem(args) = 0 ) end proc:

$A≔\mathrm{Matrix}\left(\left[\left[3\,4\,5\,7\right]\,\left[5\,7\,11\,13\right]\,\left[7\,11\,13\,17\right]\right]\right)$

$A≔\left[\begin{array}{cccc}3& 4& 5& 7\\ 5& 7& 11& 13\\ 7& 11& 13& 17\end{array}\right]$

$\mathrm{BareissAlgorithm}\left[Z\right]\left(A\,'r'\right)$

$\left[\begin{array}{cccc}3& 4& 5& 7\\ 0& 1& 8& 4\\ 0& 0& \mathrm{-12}& \mathrm{-6}\end{array}\right]$

$r$

$3$