|\^/|     Maple 2022 (X86 64 LINUX)
._|\|   |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2022
 \  MAPLE  /  All rights reserved. Maple is a trademark of
 <____ ____>  Waterloo Maple Inc.
      |       Type ? for help.
> FFGE := proc(A::Matrix)
# Bareiss/Edmonds over the integers
> local n,m,mu,det,i,j,k,N;
>    n,m := op(1,A);
>    if n<>m then error "Matrix is not square\n" fi;
>    mu := 1;
>    det := 1;
>    for k to n-1 do
>        i := k;
>        while i<=n and A[i,k]=0 do i++; od;
>        if i>n then return 0; fi;
>        if i>k then # interchange rows i and k
>            det := -det;
>            for j from k to n do
>                A[i,j],A[k,j] := A[k,j],A[i,k];
>            od;
>        fi;
>        for i from k+1 to n do
>            for j from k+1 to n do
>                N := A[k,k]*A[i,j]-A[i,k]*A[k,j];
>                A[i,j] := iquo(N,mu);
>            od;
>            A[i,k] := 0;
>        od;
>        mu := A[k,k];
>    od;
>    det;
> end:

> with(LinearAlgebra):
> c := rand(10^4):
> for n from 3 to 4 do
>     A := Matrix(n,n,c);
>     Determinant(A);
>     FFGE(A);
>     A;
> od;
                               [7926    8057       5]
                               [                    ]
                          A := [3002    2347    9765]
                               [                    ]
                               [3354    5860    6906]

                                 -228187346372

                                       1

                      [7926        8057                5]
                      [                                 ]
                      [   0    -5584792         77382380]
                      [                                 ]
                      [   0           0    -228187346372]

                           [5281    5393    1203     311]
                           [                            ]
                           [9386    9810    5144    7995]
                      A := [                            ]
                           [3121    9390    2055    6505]
                           [                            ]
                           [5293    2987    2440    8012]

                                -569192119134689

                                       1

             [5281       5393            1203                 311]
             [                                                   ]
             [   0    1187912        15874106            39302549]
             [                                                   ]
             [   0          0    -96867450578       -236277191425]
             [                                                   ]
             [   0          0               0    -569192119134689]

> quit
memory used=3.0MB, alloc=8.3MB, time=0.04