International Mathematics Competition
for University Students
2018

IMC2018: Day 1, Problem 3

Problem 3. Determine all rational numbers \(\displaystyle a\) for which the matrix

\(\displaystyle \begin{pmatrix} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a \\ 0 & 1 & a & -a \end{pmatrix} \)

is the square of a matrix with all rational entries.

(Proposed by Daniël Kroes, University of California, San Diego)

Hint: Let \(\displaystyle A = \begin{pmatrix} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a \\ 0 & 1 & a & -a \end{pmatrix}\) and suppose that \(\displaystyle A=B^2\). What can be the minimal polynomial of \(\displaystyle B\)?

  Solution