If \({a}_{r}=\cos \frac{2r\pi }{9}+i\sin \frac{2r\pi }{9},\) \(r=1,2,3,...,i=\sqrt{-1},\) then the determinant \(\left|\…

If \({a}_{r}=\cos \frac{2r\pi }{9}+i\sin \frac{2r\pi }{9},\)

\(r=1,2,3,...,i=\sqrt{-1},\) then the determinant \(\left|\begin{matrix}{a}_{1} & {a}_{2} & {a}_{3} \\ {a}_{4} & {a}_{5} & {a}_{6} \\ {a}_{7} & {a}_{8} & {a}_{9}\end{matrix}\right|\) is

[JEE Main 2021, 31 Aug (Shift 1)]