Matrix exponential of a positive matrix

Let $M$ be an $ntimes n$ real matrix with strictly positive elements, i.e. $M_{ij}>0, forall 1leq i,jleq n$, and let $lambda_1$ be its Perron-Frobenius eigenvalue (i.e. the eigenvalue with largest magnitude, must be real and positive). Is the following inequality always correct?
$$[e^{M}]_{ij}leq e^{lambda_1}, forall 1leq i,j leq n$$

It is easy to prove this when $M$ is symmetric, since when $M$ is symmetric, we have $|M|=lambda_1$, therefore
$$[e^{M}]_{ij}leq |e^{M}|leq e^{|M |} = e^{lambda_1}, forall 1leq i,j leq n.$$
When $M$ is not symmetric, I checked several cases numerically and the answer seems to be true, but I don’t know how to prove this. [The above proof fails because $|M|neq lambda_1$ when $M$ is not symmetric.]