Exercise 10, § 6

Soit A a real matrix of type $(I,I)$, positive and irreducible. For every integer $n$ such that $n\geqslant$ Card(I), the matrix $(1_I+$ A$)^{n-1}$ is strictly positive. (Let $y\in \mathbf{R}_+^n$ and $z= (1_I+$ A)$y$; if $y\not = 0$, prove that the set of the $i\in I$ such that $z_i= 0$ is strictly contained in $\{i\in I|y_i= 0\}.)$