Exercise 16, § 10

a) Let $U$ be a square matrix of order n over the field $\mathbf{Q}$, whose all 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 every i and $d_i > r$ perhaps for one value of i. Now prove 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 $x_i$ are all of the same sign and deduce that they are all zero). b) Let E be a finite set having m elements denoted by $a_i$ ($1 \leq i \leq m$), $(A_j)_{1 \leq j \leq n}$ a family of n parts of E, pairwise distinct; suppose that one has Card$(A_i \cap A_j) = r \geq 1$ for every pair (i, j) of distinct elements of {1, n}. Now prove that one necessarily has $n \leq m$. (Let $V = (c_{ij})$ be the matrix of type (m, n) such that $c_{ij} = 1$ when $a_i \in A_j, c_{ij} = 0$ in the contrary case. Apply a) to the matrix $^tV.V$ of order n.)