Do there exist $n\times n$ real  matrices  $A_{1}, \ldots, A_{n}$  such that $n$-homogeneous polynomial 
$$
f(x_{1}, \ldots, x_{n}) = \det(x_{1} A_{1}+\cdots +x_{n} A_{n})
$$ 
is never zero on $\mathbb{R}^{n}\setminus\{0\}$?

I was listening to a seminar of a student, and a problem  boils down to this linear algebraic question. I know that if $n$ is odd the answer is negative,  and if $n=2^{k}$ the answer is positive. I do not quite see right now  what happens for an arbitrary even $n$.  The first interesting case is $n=6$. 

I believe this should be something well known.