Let $A$ denote a positive matrix in $\mathcal{M}_n(\mathbb{R})$. Let $i, j$ be in $\llbracket 1, n \rrbracket$, and let $\ell$ and $m$ be in $\mathbb{N}^*$. Show the equivalence of the propositions:
\begin{itemize}
\item there exists in $A^m$ a path with origin $i$, endpoint $j$, of length $\ell$;
\item there exists in $A$ a path with origin $i$, endpoint $j$, of length $m\ell$.
\end{itemize}