IMC 2018 · Problem 3

Day 125th IMC · Blagoevgrad, Bulgaria

Statement

Determine all rational numbers $a$ for which the matrix

\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.

Hidden so you can work on the problem first.

Proposed by Daniël Kroes, University of California, San Diego.