Question

If w is an imaginary cube root of unity, find the value of $\left|\begin{array}{ccc}1& w& w^2\\ w& w^2& 1\\ w^2& 1& w\end{array}\right|$.

Answer 1

$$\begin{gathered} \left|1w{w}^2 \\ w{w}^{2}1 \\ {w}^{2}1w\right| \\ =\left|1+w+w^{2}w{w}^2 \\ w+w^2+1w^{2}1 \\ {w}^2+1+w1w\right|\left[\mathrm{Applying}{\mathrm{C}}_1\to {\mathrm{C}}_1+{\mathrm{C}}_2+{\mathrm{C}}_3\right] \\ =\left|0w{w}^2 \\ 0w^{2}1 \\ 01w\right|\left[\because 1+w+w^2=0,w\mathrm{is}\mathrm{the}\mathrm{imaginary}\mathrm{cube}\mathrm{root}\mathrm{of}\mathrm{unity}\right] \\ =0 \end{gathered}$$