Exercise 16, § 6

Let A be a real square matrix of type $(I,I)$, positive and irreducible. For every $j\in I$, put $s_j=\sum_{k\in I}a_{jk}$,

$s=$ inf$_{j\in I}s_j$, and S = sup$_{j\in I}s_j$.

Let us also adopt the notations $r$ and B(X) of exercise 12.

a) We have $\sum_{j\in I}(r-s_j)B_{kj}(r) = 0$.

b) Deduce that $s\leqslant r\leqslant S$ and that the case of equality in one of these two inequalities implies that $s= S$.