Throughout this subsection, $A$ is a given primitive matrix in $\mathcal{M}_n(\mathbb{R})$. We denote by $\ell \in \llbracket 1, n \rrbracket$ the smallest length of an elementary circuit of $A$.
By contradiction, we suppose $\ell = n$.
Show that then all circuits of $A$ have length a multiple of $n$. Deduce that the matrices $A^{kn+1}$ (with $k \in \mathbb{N}$) have zero diagonal and reach a contradiction.