Problem Solving Set 2

05 July 2012

1. (a) Let A be a n × n, n ≥ 2, symmetric, invertible matrix with real positive elements. Show that z_{n} ≤ n^{2} − 2n, where z_{n} is the number of zero elements in A^{−1}.

(b) How many zero elements are there in the inverse of the n × n matrix

(\begin{matrix}1 & & 1 & & 1 & & 1 & … & & 1 \\ 1 & & 2 & & 2 & & 2 & … & & 2 \\ 1 & & 2 & & 1 & & 1 & … & & 1 \\ 1 & & 2 & & 1 & & 2 & … & & 2 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 1 & & 2 & & 1 & & 2 & … & & … \end{matrix})?

2. Let a_{1}, a_{2}, ..., a_{n} be distinct integers. Show that

f(x) = (x − a_{1})(x − a_{2})...(x − a_{n}) − 1

is irreducible over the rationals.