Markov processes and generator

Let be a differentiable semi-goup of Markov transition matrices with the infinitesimal matrix (generator) . Assuming that , how do you prove that for every all the matrix entries of are ?

(Prove then that there is a unique stationary distribution for the semi-group of transition matrices.)

Note there is a hint: represent as with a constant c sufficiently large so that to make all the elements of the matrix non-negative.

The main difficulty comes from the diagonal elements of .

Thanks for your help.