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 numbers 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 _________.

$\mathrm{\omega }={\mathrm{e}}^{\mathrm{i}2\mathrm{\pi }/3}$

$\begin{array}{l}\left|\begin{array}{ccc}\mathrm{z}+1& \mathrm{\omega }& {\mathrm{\omega }}^2\\ \mathrm{\omega }& \mathrm{z}+{\mathrm{\omega }}^2& 1\\ {\mathrm{\omega }}^2& 1& \mathrm{z}+\mathrm{\omega }\end{array}\right|=0\\ \mathrm{z}\left|\begin{array}{ccc}1& \mathrm{\omega }& {\mathrm{\omega }}^2\\ 1& \mathrm{z}+{\mathrm{\omega }}^2& 1\\ 1& 1& \mathrm{z}+\mathrm{\omega }\end{array}\right|=0\end{array}$

$\text{ (Applying }{\mathrm{C}}_1\to {\mathrm{C}}_1+{\mathrm{C}}_2+{\mathrm{C}}_3\text{ and using }1+\mathrm{\omega }+{\mathrm{\omega }}^2=0\text{ ) }$

$\begin{array}{l}\mathrm{or} \mathrm{z}\left[\left(\mathrm{z}+{\mathrm{\omega }}^2\right)(\mathrm{z}+\mathrm{\omega })-1-\mathrm{\omega }(\mathrm{z}+\mathrm{\omega }-1)+{\mathrm{\omega }}^2\left(1-\mathrm{z}-{\mathrm{\omega }}^2\right)\right]=0\\ \mathrm{or} {\mathrm{z}}^3=0\\ \mathrm{or} \mathrm{z}=0\text{ is only solution. }\end{array}$