Order inconsistency when inverting a matrix of `Taylor1`s

[LuEdRaMo]

Suppose we have matrix of Taylor1s generated as follows and we want to compute its inverse:

using LinearAlgebra, TaylorSeries
t = Taylor1(15)
M = diagm(rand(3)) .* one(t)
inv_M = inv(M)

I would expect every element of both M and inv_M to have the same order as t; however, when running the code above I noticed that two elements of inv_M have order 0.

@lbenet Can you reproduce this? If so, is this an inconsistency?

[lbenet]

Yes, I can reproduce it just as you describe it (Julia v1.12, but I guess the same happens in other versions). And to be more specific, all non-diagonal elements are of order 0.

My guess, though I haven't checked it, is that LinearAlgebra recognizes that it is a diagonal matrix, and then only computes the diagonal elements of the inverse, filling the off diagonal elements with zero(Taylor1{Float64}) (instead of zero(M[1,1])). This produces those order-0 elements. If you try to do the same with a non-diagonal matrix, it seems that things work as expected.