On the topological properties and measure of $\{(A,B)\in M(n,\mathbb R)\times M(n,\mathbb R) : \det (A+B)=\det(A)+\det(B)\}$

Question

For $n\ge 2$ , let $T(n):=\{(A,B)\in M(n,\mathbb R)\times M(n,\mathbb R) : \det (A+B)=\det(A)+\det(B)\}$ .

I can see that $T(n)$ is closed and non-compact in the usual norm of $M(n,\mathbb R)^2 \cong \mathbb R^{2n^2}$ . My questions are :

(1) Is $T(n)$ connected or path connected ?

(2) Is the Lebesgue measure of $T(n)$ zero ?

(3) Is the interior of $T(n)$ empty ?

Answer 1

Perhaps a bit long for a comment but not very deep: the anser to 2 is yes, from which it follows that the answer to 3 is yes as well. For 2: it is not just closed in the ordinary topology, but even in the Zariski-toplogy: it is the zero-set of a polynomial function. The only way that can have positive measure is if it is the enitre space, so it suffices to show that there is a point $(A, B)$ not of this form. This is not particularly hard.

Question 1 is interesting, though!