If $\mathrm{abc}=1$ and $A=\left[\begin{array}{lll}a & b & c \ c & a & b \ b & c & a\end{array} i

Question

If $\mathrm{abc}=1$ and $A=\left[\begin{array}{lll}a & b & c \ c & a & b \ b & c & a\end{array}ight]$ is an orthogonal matrix then

Professor PC Thomas Classes · Answer

$ \begin{aligned} & a^2+b^2+c^2=1, a b+b c+c a=0 \ & \Rightarrow(a+b+c)^2=1 \Rightarrow a+b+c= \pm 1 \ & a^3+b^3+c^3-3 a b c \ & =(a+b+c)\left(a^2+b^2+c^2-a b-b c-c aight) \ & = \pm 1 \ & \Rightarrow a^3+b^3+c^3= \pm 1+3=4 	ext { or } 2 \end{aligned} $