TITLE: Matrix exponential: $e^A\ge 0\iff a_{ij}\ge 0$ when $i\neq j$
QUESTION [8 upvotes]: Let $A$ be a $n$ by $n$ matrix. Prove that that $$a_{ij}\ge 0 \text{ whenever }i\neq j\iff e^A\text{ has all entries }\ge 0.$$ 
I'd like just a hint for now please.

REPLY [7 votes]: There's already a counter-example by user1551, but here's an even simpler one.
If we let
$$A=\left[\matrix{0&1\\-1&0}\right],$$
then $e^{\theta A}$ rotates the plane by an angle $\theta$. Enter $\theta=2\pi$ and you get $e^{2\pi A}=I$ where $I$ is the identity matrix which has non-negative elements.