In their paper "User's guide for viscosity solutions of second order partial differential equations", Crandall, Ishii and Lion claim that
$F(x, r, p, \mathcal X) = - tr(A(x)\mathcal X) + \sum_i b_i(x) p_i(x) + c(x)r - f(x)$ is degenerate elliptic if and only if $A(x) \geq 0$,
where $F: \mathbb R^N \times \mathbb R \times \mathbb R^N \times \mathcal S_N \longrightarrow \mathbb R$, $\mathcal S_N$ is the set of real symmetric matrices of $N$ lines and $A(x) \in \mathcal S_N$. $F$ is said to be degenerate elliptic if $$ \mathcal X \geq \mathcal Y \implies F(x, r, p, \mathcal X) \leq F(x, r, p, \mathcal Y), $$ where $\mathcal X \geq \mathcal Y$ means that $\mathcal X - \mathcal Y$ is positive semidefinite.
If $A$ is positive semidefinite, it is easy to see that $F$ is degenerate elliptic. What about the other direction (namely, the statement in the title)?
Using that $I \geq 0$ we obtain that $tr(A) \geq 0$, but this doesn't seem to help.
Thanks in advance.