|\^/|     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)
> local n,m,mu,det,i,j,k,N,Q;
>    n,m := op(1,A);
>    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
>            det := -det;
>            for j from k to n do
>                A[i,j],A[k,j] := A[k,j],A[i,j];
>            od;
>        fi;
>        for i from k+1 to n do
>            for j from k+1 to n do
>                N := expand(A[k,k]*A[i,j]-A[i,k]*A[k,j]);
>                divide(N,mu,'Q');
>        if k = n-1 then # last step
>            printf("k=%d i=%d j=%d #Ak-1k-1=%d #N=%d #Q=%d #N/#Q=%.3f\n",
>            k,i,j,numterms(mu),numterms(N),numterms(Q),
>            evalf(numterms(N)/numterms(Q))); 
>        fi;
>                A[i,j] := Q;
>            od;
>            A[i,k] := 0;
>        od;
>        mu := A[k,k];
>    od;
>    A[n,n];
> end:
>
> GetSymMat := proc(n,x)
> local A,i,j,k;
>     A := Matrix(n,n);
>     k := 1;
>     for i to n do
>         A[i,i] := x[k]; k++;
>         for j from i+1 to n do 
>             A[i,j] := x[k];
>             A[j,i] := x[k];
>             k++;
>         od;
>     od;
>     A;
> end:

> numterms := proc(f) 
>    if f=0 then 0 elif type(f,`+`) then nops(f) else 1 fi; 
> end:

> with(LinearAlgebra):

# Test
> S3 := GetSymMat(3,x);
                                [x[1]    x[2]    x[3]]
                                [                    ]
                          S3 := [x[2]    x[4]    x[5]]
                                [                    ]
                                [x[3]    x[5]    x[6]]

> det := Determinant(S3);
                                 2       2                               2
det := x[1] x[4] x[6] - x[1] x[5]  - x[2]  x[6] + 2 x[2] x[3] x[5] - x[3]  x[4]

> FFGE(S3);
k=2 i=3 j=3 #Ak-1k-1=1 #N=5 #Q=5 #N/#Q=1.000
                              2       2                               2
    x[1] x[4] x[6] - x[1] x[5]  - x[2]  x[6] + 2 x[2] x[3] x[5] - x[3]  x[4]

> S3;
    [x[1] , x[2] , x[3]]

    [                    2                        ]
    [0 , x[1] x[4] - x[2]  , x[1] x[5] - x[2] x[3]]

    [
    [0 , 0 ,

                              2       2                               2     ]
    x[1] x[4] x[6] - x[1] x[5]  - x[2]  x[6] + 2 x[2] x[3] x[5] - x[3]  x[4]]


> for n from 3 to 8 do
>    A := GetSymMat(n,x);
>    det1 := Determinant(A);
>    det2 := FFGE(A);
>    print(n,evalb(det1=det2),numterms(det1));
> od:
k=2 i=3 j=3 #Ak-1k-1=1 #N=5 #Q=5 #N/#Q=1.000
                                   3, true, 5

k=3 i=4 j=4 #Ak-1k-1=2 #N=32 #Q=17 #N/#Q=1.882
                                  4, true, 17

k=4 i=5 j=5 #Ak-1k-1=5 #N=324 #Q=73 #N/#Q=4.438
                                  5, true, 73

memory used=3.4MB, alloc=40.3MB, time=0.05
k=5 i=6 j=6 #Ak-1k-1=17 #N=5292 #Q=388 #N/#Q=13.639
                                  6, true, 388

memory used=60.9MB, alloc=82.5MB, time=0.42
k=6 i=7 j=7 #Ak-1k-1=73 #N=123432 #Q=2461 #N/#Q=50.155
                                 7, true, 2461

memory used=145.6MB, alloc=84.7MB, time=1.05
memory used=226.5MB, alloc=186.3MB, time=1.79
memory used=434.5MB, alloc=250.3MB, time=3.89
memory used=588.5MB, alloc=346.3MB, time=5.16
memory used=665.2MB, alloc=410.3MB, time=6.66
memory used=690.1MB, alloc=442.3MB, time=8.44
memory used=845.6MB, alloc=570.3MB, time=11.76
memory used=957.7MB, alloc=698.3MB, time=16.39
memory used=1109.7MB, alloc=858.3MB, time=22.24
memory used=1529.6MB, alloc=1165.1MB, time=33.90
memory used=1791.8MB, alloc=1389.1MB, time=46.41
memory used=2445.7MB, alloc=1775.3MB, time=71.02
k=7 i=8 j=8 #Ak-1k-1=388 #N=3933630 #Q=18155 #N/#Q=216.669
                                 8, true, 18155

So the maximum expression swell at n=8 is a factor of 217