Let $\mathrm{\omega }$ be the complex number $\cos \frac{2\mathrm{\pi }}{3}+\mathrm{isin}\frac{2\mathrm{\pi }}{3}$. Then the number of distinct complex number z satisfying $\left|\begin{array}{ccc}\mathrm{z}+1& \mathrm{\omega }& {\mathrm{\omega }}^2\\ \mathrm{\omega }& \mathrm{z}+{\mathrm{\omega }}^2& 1\\ {\mathrm{\omega }}^2& 1& \mathrm{z}+\mathrm{\omega }\end{array}\right|=0$ is equal to  

$\begin{array}{l}\mathrm{\omega }={\mathrm{e}}^{\mathrm{i}2\mathrm{\pi }/3}\\ \left|\begin{array}{ccc}\mathrm{z}+1& \mathrm{\omega }& {\mathrm{\omega }}^2\\ \mathrm{\omega }& \mathrm{z}+{\mathrm{\omega }}^2& 1\\ {\mathrm{\omega }}^2& 1& \mathrm{z}+\mathrm{\omega }\end{array}\right|=0\\ {\mathrm{C}}_1\to {\mathrm{C}}_1+{\mathrm{C}}_2+{\mathrm{C}}_3\end{array}$

$$\begin{gathered} \left|\begin{array}{ccc}z& \omega & {\omega }^2\\ z& z+{\omega }^2& 1\\ z& 1& z+\omega \end{array}\right|=0 \\ \Rightarrow \mathrm{z}\left|\begin{array}{ccc}1& \omega & {\omega }^2\\ 1& \mathrm{z}+{\omega }^2& 1\\ 1& 1& z+\omega \end{array}\right|=0 \\ {\mathrm{R}}_2\to {\mathrm{R}}_2-{\mathrm{R}}_1,{\mathrm{R}}_3\to {\mathrm{R}}_3-{\mathrm{R}}_1 \\ \Rightarrow \mathrm{z}\left|\begin{array}{ccc}1& \omega & {\omega }^2\\ 0& \mathrm{z}+{\omega }^2-\omega & 1-{\omega }^2\\ 0& 1-\omega & \mathrm{z}+\omega -{\omega }^2\end{array}\right|=0 \\ =\left(\mathrm{z}+{\omega }^2-\omega \right)\left(\mathrm{z}+\omega -{\omega }^2\right)-(1-\omega )\left(1-{\omega }^2\right)=0 \\ z=0 \\ one solution \end{gathered}$$