for non-trivial solution $$\left | A \right | = 0$$
where $\left | A \right | = \begin{bmatrix} p & q& r\\ q& r& p\\ r& p & q \end{bmatrix} = p*(rq-p^{2})-q*(q^{2}-pr)+r*(qp-r^{2})$
$=prq - p^3 - q^3 + prq + prq - r^3 \\= 3prq - p^3 - q^3 - r^3 \\=-{\left(p+q+r\right)}^3 + 3(p+q+r)(pq+qr+pr)$
now if you check the options the only options where each individual condition can make $\left | A \right | = 0$ zero is C