Question

If $\omega$ be imaginary root of unity then evaluate $\left|\begin{array}{ccc}2+3{\omega }^{100}+{\omega }^{203}& 2{\omega }^2& 1\\ 1& 1+{\omega }^{100}+3{\omega }^{203}& 2\omega \\ 2\omega & 2{\omega }^2& 3+{\omega }^{100}+{\omega }^{203}\end{array}\right|$

Answer 1

${\omega }^{100}=({\omega }^3{)}^{33}.\omega =\omega ,{\omega }^{203}=({\omega }^{100}{)}^2.{\omega }^3$
=${\omega }^2.1={\omega }^2$
Apply $C_1+C_2+C_3$ then new $C_1$ becomes a column of zeros as $3(1+\omega +{\omega }^2)=0$