Exercise 16, § 10

(a) Let $U$ be a square matrix of order $n$ over the field $\mathbf{Q}$ all of whose non-diagonal elements are equal to the same rational number $r > 0$ and whose diagonal $(d_1, \ldots, d_n)$ is such that $d_i \geq r$ for all $i$ and $d_i > r$ except perhaps for one value of $i$. Show that $U$ is invertible (if $x = (x_i)_{1 \leq i \leq n}$ is a solution of the equation $U.x = 0$, show that the $x_i$ all have the same sign and deduce that they are all zero).

(b) Let $E$ be a finite set with $m$ elements denoted by $a_i$ ($1 \leq i \leq m$) and $(A_j)_{1 \leq j \leq n}$ a family of $n$ subsets of $E$, distinct from one another; suppose that $\operatorname{Card}(A_i \cap A_j) = r \geq 1$ for every ordered pair $(i, j)$ of distinct elements of $\{1, n\}$. Show that necessarily $n \leq m$. (Let $V = (c_{ij})$ be the matrix of type $(m, n)$ such that $c_{ij} = 1$ when $a_i \in A_j, c_{ij} = 0$ otherwise. Apply (a) to the matrix $tV.V$ of order $n$.)