IMC 2018 Problem 3

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.

Proposed by Daniël Kroes, University of California, San Diego