Evaluate the cyclic determinant $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$.
Step-by-Step Solution
Key Concept: General
$$\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = -(a^3 + b^3 + c^3 - 3abc) = -(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ac) = -\frac{1}{2}(a + b + c)\{(a - b)^2 + (b - c)^2 + (c - a)^2\} = -(a + b + c)(a + b\omega + c\omega^2)(a + b\omega^2 + c\omega)$$, where $\omega, \omega^2$ are cube roots of unity.
Correct Answer: -(a^3 + b^3 + c^3 - 3abc)