zeros of sum of two (matrix-valued!) polynomials

hi,

I have a nontrivial problem. let f(z) and g(z) be two (possibly monic) matrix-valued polynomials (i.e. of the form $\displaystyle \sum_{j=0}^n A_j z^j$ with $\displaystyle A_j$ being $\displaystyle m\times m$ matrices, and $\displaystyle z\in\mathbb{C}$). assume that det(f(z)) and det(g(z)) has all real and negative roots. assume det(f(z)+g(z)) has all roots real.

prove (or disprove) that det(f(z)+g(z)) cannot have a positive zero.

for the case of dimension 1, it's easy since f(z) and g(z) are positive for z>0, and so f(z)+g(z) is positive for z>0, so all the roots are negative.

any suggestion/idea/reference would be highly appreciated...