using LinearAlgebra

julia> A = I(2)
2×2 Diagonal{Bool, Vector{Bool}}:
 1  ⋅
 ⋅  1

julia> inv(A) # correct, diagonal matrix
2×2 Diagonal{Float64, Vector{Float64}}:
 1.0   ⋅ 
  ⋅   1.0

julia> inv(Symmetric(A)) # dense matrix?
2×2 Matrix{Float64}:
 1.0  0.0
 0.0  1.0

inv(A::Symmetric{<:Any, <:Diagonal}) = inv(A.data)