Question:
If $w$ is an imaginary cube root of unity, find the value of $\left|\begin{array}{ccc}1 & w & w^{2} \\ w & w^{2} & 1 \\ w^{2} & 1 & w\end{array}\right|$.
Solution:
$\mid \begin{array}{lll}1 & w & w^{2}\end{array}$
$w \quad w^{2} \quad 1$
$w^{2} \quad 1 \quad w \mid$
$=\mid 1+w+w^{2} \quad w \quad w^{2}$
$w+w^{2}+1 \quad w^{2} \quad 1$
$w^{2}+1+w \quad 1 \quad w \mid$
$\left[\right.$ 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 $]$
$=0$