Let $A=\begin{bmatrix}p & q & r\\q & r & p\\r & p & q \end{bmatrix}$.

Since the system has non-trivial solution, hence

$\begin{align} |A|&=0\\ \Rightarrow \left|\begin{matrix}p & q & r\\q & r & p\\r & p & q \end{matrix}\right| &=0\\ \Rightarrow \left|\begin{matrix}p+q+r & q & r\\p+q+r & r & p\\p+q+r & p & q \end{matrix}\right| &=0;~[C_1 \leftarrow C_1+C_2+C_3]\\ \Rightarrow (p+q+r)\left|\begin{matrix}1 & q & r\\1 & r & p\\1 & p & q \end{matrix}\right| &=0\\ \Rightarrow (p+q+r)\left\{ (qr-p^2) -(q^2-pr)+(pq-r^2) \right\} &=0; ~[\scriptsize\text{Expanding the determinant using the 1st column}]\\ \Rightarrow (p+q+r)(p^2+q^2+r^2-pq-qr-rp) &=0 \end{align}$

$\therefore p+q+r=0\mathrm{~Or~}(p^2+q^2+r^2-pq-qr-rp)=0~...............\tag{i}$

Now

$\begin{align} p^2+q^2+r^2-pq-qr-rp&=0\\ \Rightarrow 2p^2+2q^2+2r^2-2pq-2qr-2rp &=0;~[\scriptsize\text{Multiplying both sides by }2]\\ \Rightarrow (p^2-2pq+q^2)+(q^2-2qr+r^2)+(r^2-2rp+p^2) &=0;~[\scriptsize\text{Rearranging the expression}]\\ \Rightarrow (p-q)^2+(q-r)^2+(r-p)^2 &=0\\ \therefore p-q=0 \mathrm{~And~}q-r=0 \mathrm{~And~}r-p&=0;~[\scriptsize\text{Since the sum of some squares is }0 \text{, hence they must be separately }0.] \\ \Rightarrow p=q \mathrm{~And~}q=r \mathrm{~And~}r=p\\ \Rightarrow p=q=r \end{align}$

Therefore, from no$\mathrm{(i)}$, we can conclude that

$$p+q+r=0 \mathrm{~Or~} p=q=r$$.

So the correct answer is **C**.