$z = \sqrt {3i} $

$ \Rightarrow \omega  = \frac{{\sqrt {3i}  - 1}}{2}$

$ \Rightarrow \omega $ is complex cube root of unity 

Applying ${R_1} \to {R_1} + {R_2} + {R_3}$

$ = \left| \begin{array}{l}
3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\\
1\,\,\,\,\, - {\omega ^2} - 1\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^2}\\
1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\omega 
\end{array} \right|\,$

$ = 3\left( { - 1 - \omega  - \omega } \right) =  - 3\left( {1 + 2\omega } \right)\, =  - 3z$

$ \Rightarrow k =  - z$