Fix eltype determination in det (#44582)

Co-authored-by: Andreas Noack <andreas@noack.dk>

stdlib/LinearAlgebra/src/generic.jl

@@ -1539,9 +1539,9 @@ julia> det(M)
 2.0
 ```
 """
-function det(A::AbstractMatrix{T}) where T
+function det(A::AbstractMatrix{T}) where {T}
     if istriu(A) || istril(A)
-        S = typeof((one(T)*zero(T) + zero(T))/one(T))
+        S = promote_type(T, typeof((one(T)*zero(T) + zero(T))/one(T)))
         return convert(S, det(UpperTriangular(A)))
     end
     return det(lu(A; check = false))

stdlib/LinearAlgebra/test/generic.jl

@@ -76,6 +76,35 @@ n = 5 # should be odd
         @test logabsdet(x)[1] ≈ logabsdet(X)[1]
         @test logabsdet(x)[2] ≈ logabsdet(X)[2]
     end
+
+    @testset "det with nonstandard Number type" begin
+        struct MyDual{T<:Real} <: Real
+            val::T
+            eps::T
+        end
+        Base.:+(x::MyDual, y::MyDual) = MyDual(x.val + y.val, x.eps + y.eps)
+        Base.:*(x::MyDual, y::MyDual) = MyDual(x.val * y.val, x.eps * y.val + y.eps * x.val)
+        Base.:/(x::MyDual, y::MyDual) = x.val / y.val
+        Base.:(==)(x::MyDual, y::MyDual) = x.val == y.val && x.eps == y.eps
+        Base.zero(::MyDual{T}) where {T} = MyDual(zero(T), zero(T))
+        Base.zero(::Type{MyDual{T}}) where {T} = MyDual(zero(T), zero(T))
+        Base.one(::MyDual{T}) where {T} = MyDual(one(T), zero(T))
+        Base.one(::Type{MyDual{T}}) where {T} = MyDual(one(T), zero(T))
+        # the following line is required for BigFloat, IDK why it doesn't work via
+        # promote_rule like for all other types
+        Base.promote_type(::Type{MyDual{BigFloat}}, ::Type{BigFloat}) = MyDual{BigFloat}
+        Base.promote_rule(::Type{MyDual{T}}, ::Type{S}) where {T,S<:Real} =
+            MyDual{promote_type(T, S)}
+        Base.promote_rule(::Type{MyDual{T}}, ::Type{MyDual{S}}) where {T,S} =
+            MyDual{promote_type(T, S)}
+        Base.convert(::Type{MyDual{T}}, x::MyDual) where {T} =
+            MyDual(convert(T, x.val), convert(T, x.eps))
+        if elty <: Real
+            @show elty
+            @show istriu(triu(MyDual.(A, zero(A))))
+            @test det(triu(MyDual.(A, zero(A)))) isa MyDual
+        end
+    end        
 end
 
 @testset "diag" begin