If p, q, r are the cube roots of unity, then what is |p²+q² r² r² ; p² q²+r² p² ; q² q² r²+p²| equal to?

The cube roots of unity satisfy p + q + r = 0 and p² + q² + r² = (p+q+r)² − 2(pq+qr+rp) = 0 − 2(0) = 0 (since pq+qr+rp = 0 for cube roots of unity). Adding all three columns of the determinant: each row sum = 2(p² + q² + r²) = 0. So the determinant equals 0.