If w is an imaginary cube root of unity,

$\mid \begin{array}{lll}1 & w & w^{2}\end{array}$

$w \quad w^{2} \quad 1$

$w^{2} \quad 1 \quad w \mid$

$=\mid 1+w+w^{2} \quad w \quad w^{2}$

$w+w^{2}+1 \quad w^{2} \quad 1$

$w^{2}+1+w \quad 1 \quad w \mid$

$\left[\right.$ Applying $\left.\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}+\mathrm{C}_{3}\right]$

$=\mid \begin{array}{lll}0 & w & w^{2}\end{array}$

$\begin{array}{lll}0 & w^{2} & 1\end{array}$

$\begin{array}{lll}0 & 1 & w \mid\end{array}$         $\left[\because 1+w+w^{2}=0, w\right.$ is the imaginary cube root of unity $]$

$=0$