Is this matrix exponential strictly positive?

Question

Is it true that $e^{Nt}$ is strictly positive element-wise, where $t>0$ and $N$ is a real matrix with $N_{ij}\geq0$ if $i\neq j$ and $N_{ij}\leq 0$ if $i=j$?

Answer 1

No this is not true

Take for a counterexample

$$N = \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}$$