Fixed issue #542.

[IvanYashchuk]

Added tracking of LinearAlgebra.det and its grad method.

The gradient of the determinant is taken from
http://www.cs.cmu.edu/~zkolter/course/15-884/linalg-review.pdf (Section 4.5)

It is also mentioned that $det{A} A^{-T}$ can be used, this form is also written in
Matrix Cookbook Sec. 2.1.2
https://www.ics.uci.edu/~welling/teaching/KernelsICS273B/MatrixCookBook.pdf

I don't know what variant is more appropriate.

[KristofferC]

Regarding your first implementation, note that adj here means Adjugate matrix and not adjoint (and in fact adj(A) = 1/det(A) * inv(A)

[IvanYashchuk]

Yes, when I thought about it again, I noticed it.
I've changed it already.

[IvanYashchuk]

Can someone guide me, how should it be tested?

The result matches ForwardDiff.

julia> f(x) = det(x)
julia> df(x) = Flux.Tracker.gradient(f, x)[1]
julia> a = rand(3, 3)
3×3 Array{Float64,2}:
 0.92519   0.494903   0.380887
 0.688972  0.0461206  0.226083
 0.961183  0.358504   0.60728
julia> ForwardDiff.gradient(f, a) -  df(a)
Tracked 3×3 Array{Float64,2}:
  0.0          -5.55112e-17   8.32667e-17
 -2.77556e-17   2.77556e-17   0.0
 -4.16334e-17   2.77556e-17  -5.55112e-17

[mcabbott]

For tests I think you just need to add this to test/tracker.jl:

@test gradtest(det, (4,4))

May I suggest adding also logdet and logabsdet at the same time? (Just Δ * transpose(inv(xs).)

Is it possible to cache the forward result? I was going to suggest the following, but it breaks second derivatives:

@grad function det(xs) 
    d = det(data(xs))
    d, Δ -> (Δ * d * transpose(inv(xs)),)
end

Note that both det and inv first work out lu(x) so even more re-use might be possible, but would add a lot of complexity.

[MikeInnes]

This is great. Thanks a lot for the patch!