If the following system has non-trivial solution, $$px + qy + rz = 0$$ $$qx + ry + pz = 0$$ $$rx + py + qz = 0,$$ then which one of the following options is TRUE?

A homogeneous linear system $AX = 0$ has a non-trivial solution if and only if the determinant of the coefficient matrix $A$ is zero:

Expanding the circulant determinant:

Using the standard algebraic identity:

For real values of $p, q, r$, this product is zero when either:

1. $p + q + r = 0$, or

2. $(p - q)^2 + (q - r)^2 + (r - p)^2 = 0 \implies p = q = r$.

Hence, the correct condition is $p + q + r = 0$ or $p = q = r$, which corresponds to option (C).