Question

If $\omega$ is a cube root of unity, then the value of determinant $\left|\begin{array}{ccc}1+\omega & {\omega }^2& \omega \\ {\omega }^2+\omega & -\omega & {\omega }^2\\ 1+{\omega }^2& \omega & {\omega }^2\end{array}\right|$ is equal to

• A. $-1+\omega$
• B. $1-\omega$
• C. $0$
• D. ${\omega }^2$

Answer 1

The correct option is A $-1+\omega$

$\left|\begin{array}{ccc}1+\omega & {\omega }^2& \omega \\ {\omega }^2+\omega & -\omega & {\omega }^2\\ 1+{\omega }^2& \omega & {\omega }^2\end{array}\right|$

$C_1\to C_1+C_2$

$\left|\begin{array}{ccc}0& {\omega }^2& \omega \\ {\omega }^2& -\omega & {\omega }^2\\ 0& \omega & {\omega }^2\end{array}\right|=0-{\omega }^2({\omega }^4-{\omega }^2)+0=-{\omega }^3({\omega }^3-\omega )$

$=-1(1-\omega )=-1+\omega$