$\text{ If }P=\left[\begin{array}{lll}a& b& c\\ c& a& b\\ b& c& a\end{array}\right]\text{ is an orthogonal matrix, then the sum of possible value(s) of }$$(a+b+c)\text{ is }$

$P{P}^T=\left[\begin{array}{lll}a& b& c\\ c& a& b\\ b& c& a\end{array}\right]\left[\begin{array}{lll}a& c& b\\ b& a& c\\ c& b& a\end{array}\right]=\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$\Rightarrow \left[\begin{array}{ccc}{a}^2+b^2+c^2& ab+bc+ca& ab+bc+ca\\ ab+bc+ca& a^2+b^2+c^2& ab+bc+ca\\ ab+bc+ca& ab+bc+ca& a^2+b^2+c^2\end{array}\right]=\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$$\begin{gathered} \Rightarrow a^2+b^2+c^2=1 and ab+bc+ca=0 \\ \Rightarrow {\left(a+b+c\right)}^2=1\Rightarrow a+b+c=\pm 1 \end{gathered}$$