$\therefore \Delta  = \left| {\begin{array}{*{20}{c}}
{y + 1}&\omega &{{\omega ^2}}\\
\omega &{y + {\omega ^2}}&1\\
{{\omega ^2}}&1&{y + \omega }
\end{array}} \right|$

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

$ \Rightarrow \Delta  = \left| {\begin{array}{*{20}{c}}
1&1&1\\
\omega &{y + {\omega ^2}}&1\\
{{\omega ^2}}&1&{y + \omega }
\end{array}} \right|$

Expanding along ${R_1}$, we get

$\Delta  = y.{y^2} \Rightarrow D = {y^3}$

Or

If $\alpha  = {\omega ^2},\beta  = \omega $ we get same value or on expansion using $\alpha  + \beta  =  - 1,\alpha \beta  = 1$ we get value ${y^3}$.