A fraction-free elimination (Bareiss) would raise the determinant's practical size ceiling from 10x10

[Rafael-SOWNet]

Split out of #992 / #998, which fixes the correctness half and names this as the part it deliberately does not do.

Where things stand after #998

Matrix.Determinant computed by Gaussian elimination left the pivots as literal divisions, so the expression was NaN wherever a pivot vanished. #998 replaces it with Laplace expansion, which never divides and so needs no condition. That is a strict improvement on both axes — the returned expression is smaller at every size, and numeric matrices got much faster, because the elimination built a symbolic quotient that InnerSimplified then had to grind down.

Measured, property only, both arms built from source on one machine:

entries	Gaussian complexity	Laplace complexity	Gaussian, numeric	Laplace, numeric
2×2	13	9
3×3	79	37
4×4	443	163
5×5	2461	833
6×6	13673	5021	164 ms	126 ms
7×7		35173
8×8			over 120 s	358 ms
10×10			over 120 s	4478 ms
11×11			over 240 s	over 240 s

What is left

Laplace is O(n!). For a fully symbolic matrix that is not an overhead — the determinant genuinely has n! terms, so it is the size of the answer, and no algorithm returns it smaller in expanded form. For a numeric or mostly-numeric matrix it is pure waste: the answer is one number, and O(n^3) work suffices.

The ceiling that leaves:

• numeric: 10×10 returns in 4.5 s, 11×11 does not return in 4 minutes
• symbolic: 7×7 returns in 142 ms with 35173 nodes; beyond that the answer itself is the problem

The algorithm

Bareiss' fraction-free elimination is O(n^3) and every division it performs is exact — the divisor provably divides the dividend, so over an integral domain the intermediate entries stay in the ring and no condition is ever introduced. That is the property #992 was actually about; the current elimination's fault was not that it divides but that its divisions do not cancel.

The wrinkle for AngouriMath is that exactness is a fact about the ring, not something Entity division notices: (a * b) / a stays written as a quotient unless something cancels it. So a Bareiss implementation over Entity has to reduce at each step and be measured, not assumed, to come out condition-free. That is exactly the sort of thing that should be checked with boundcheck and a substitution sweep rather than argued.

Suggested shape

• Bareiss for the numeric and low-symbol cases, where O(n^3) is the whole point
• Laplace kept for the small fully symbolic cases, where it is already optimal and returns in milliseconds
• the switch decided by something measured, not by n alone — a 12×12 with two variables and a 4×4 with sixteen are different problems

Not urgent: #998 already makes every size that previously worked work better, and adds three usable sizes on top. This is about extending the range, not repairing it.