Question

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

• A. $-2\omega$
• B. $-3{\omega }^2$
• C. $-1$
• D. $0$

Answer 1

The correct option is B $-3{\omega }^2$

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

$=\left|\begin{array}{ccc}0& {\omega }^2& -\omega \\ 0& \omega & -{\omega }^2\\ -1+\omega & \omega & -{\omega }^2\end{array}\right|$ $[\because C_1\to C_1+C_2]$

$=(-1+\omega )(-{\omega }^4+{\omega }^2)=(\omega -1)({\omega }^2-\omega )$
$={\omega }^3-{\omega }^2-{\omega }^2+\omega =-3{\omega }^2$