The value of the determinant $A=\left|\begin{array}{ccc}3& 3& 3\\ 2& \omega & {\omega }^2\\ 1& {\omega }^2& \omega \end{array}\right|$ where, $\omega$ is a complex cube root of unity is

Given determinant $=3\left|\begin{array}{ccc}1& 1& 1\\ 2& \omega & {\omega }^2\\ 1& {\omega }^2& \omega \end{array}\right|$

$\begin{array}{l}=3\left|\begin{array}{ccc}4& 0& 0\\ 2& \omega & {\omega }^2\\ 1& {\omega }^2& \omega \end{array}\right|{\mathrm{R}}_1\to {\mathrm{R}}_1+{\mathrm{R}}_2+{\mathrm{R}}_3\\ =12\left|\begin{array}{cc}\omega & {\omega }^2\\ {\omega }^2& \omega \end{array}\right|\\ =12\left({\omega }^2-\omega \right)\\ =12\left({\omega }^2+\omega -2\omega \right)=12(-1-2\omega )\\ =12\left\{-1-2\left(\frac{-1-\mathrm{i}\sqrt{3}}{2}\right)\right\}\end{array}$

OR $12\left\{-1-2\left(\frac{-1+i\sqrt{3}}{2}\right)\right\}$

$=12\mathrm{i}\sqrt{3}\text{ or }-12\mathrm{i}\sqrt{3}$