Problem Solving Set 12

15 July 2012

1. Let n > 1 be an odd positive integer and A = (a_{ij})_{i,j=1,...,n} be the n × n matrix with

a_{ij} = \begin{cases}2 & if i = j \\ 1 & if i − j = ±2 (mod n) \\ 0 & otherwise \end{cases}

Find det A.

2. Show that

x^{y} + y^{x} > 1

for all real x, y > 0