If omega is an imaginary root of unity,then t | vedclass

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(C) Let $\Delta = \left| \begin{array}{ccc} a & b\omega^2 & a\omega \ b\omega & c & b\omega^2 \ c\omega^2 & a\omega & c \end{array} ight|$.Applyin

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(C) Let $\Delta = \left| \begin{array}{ccc} a & b\omega^2 & a\omega \ b\omega & c & b\omega^2 \ c\omega^2 & a\omega & c \end{array} ight|$.Applying the column transformation $C_1 	o C_1 + C_2 + C_3$ is not directly helpful,so let us evaluate the determinant directly.Expanding along the first row:$\Delta = a(c^2 - ab\omega^3) - b\omega^2(bc\omega - bc\omega^4) + a\omega(ab\omega^2 - c^2\omega^2)$Since $\omega^3 = 1$ and $\omega^4 = \omega$:$\Delta = a(c^2 - ab) - b\omega^2(bc\omega - bc\omega) + a\omega(ab\omega^2 - c^2\omega^2)$$\Delta = a(c^2 - ab) - b\omega^2(0) + a\omega(ab\omega^2 - c^2\omega^2)$$\Delta = ac^2 - a^2b + a^2b\omega^3 - ac^2\omega^3$Since $\omega^3 = 1$:$\Delta = ac^2 - a^2b + a^2b - ac^2 = 0$.

If $\omega$ is an imaginary root of unity,then the value of $\left| \begin{array}{ccc} a & b\omega^2 & a\omega \\ b\omega & c & b\omega^2 \\ c\omega^2 & a\omega & c \end{array} \right|$ is

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