$a^{(m+n)}_{ij} \geq a^{(m)}_{ik}a^{(n)}_{kl}$ for non-negative Matrix $A$

Question

Let $A$ be a non-negative infinite Matrix (all entries $\geq 0$). $a_{ij}^{(n)}$ denotes the $ij$-th entry of $A^n$. Does the following inequality hold: $a^{(m+n)}_{ij} \geq a^{(m)}_{ik}a^{(n)}_{kl},\forall i,j,k,\forall m,n \in \mathbb{N}$?

We clearly have $a^{(2)}_{ij} \geq a^{(1)}_{ik}a^{(1)}_{kj},\forall i,k,j$. Can the rest be shown by a induction argument or how to make it precise?

Answer 1

In this case, every entry is positive, $$ a_{ij}^{(n+m)} = \sum_{k=1}^d a_{ik}^{(m)} a_{kj}^{(n)} \ge a_{ik_0}^{(m)} a_{k_0j}^{(n)} $$for each $k_0$. Otherwise there is no reason.