Rules for det are not general and likely unstable

A = [1.0 1.0; 1.0 1.0] 
det(A) * inv(A) 
# ERROR: SingularException(2)

to be fair, the LU approach is remarkably resiliant...

A = [1.0 1.0; 1.0 1.0-1e-16]
det(A) * inv(A)
# 2×2 Matrix{Float64}:
#   1.0  -1.0
#  -1.0   1.0

But I am nervous that this is too simple a special case to draw any conclusions.

Either way - it seems to me that an implementation that always works and is guaranteed numerically stable is preferrable. If the maintainers agree, then I would be happy to make a PR and implement frule and rrule via SVD instead of LU.

Questions:

• should the det evaluation then also use SVD, or should it use whichever the standard implementation is?
• should the SVD implementation be only for a specific class of matrices, or general? E.g. I'm unsure about sparse.
• I noticed that there are rules for svd - I'm guessing this means one will be able to differentate the frule or rrule based on SVD?

Start by reviewing the existing rules for det, inv, frule, rrule, and svd, then compare the LU and SVD approaches described in the issue. Before implementation, resolve whether det should use SVD, whether the rule should cover sparse matrices, and what numerical-stability guarantee is required; done means an agreed general design is implemented and its behavior is validated.