If $x, y, z$ distinct common roots of $z^6 - 1 = 0$ and $z^3 - 1 = 0$ then $$\begin{vmatrix} x - y - z & 2x & 2x \\ 2y & y - x - z & 2y \\ 2z & 2z & z - x - y \end{vmatrix}$$ is equal to _____.
Step-by-Step Solution
Key Concept: The common roots of polynomials $z^6 - 1$ and $z^{21} - 1$ are exactly the GCD of their orders, and row operations on determinants preserve the determinant value.
Given $z^6 - 1 = 0$ and $z^{21} - 1 = 0$, the common roots are the 6th roots of unity: $1, \omega, \omega^2$ where $\omega = e^{2\pi i/3}$. From $R_1 = R_1 + R_2 + R_3$ and the determinant equation $\begin{vmatrix} x+y+z & x+y+z & x+y+z \\ 2y & y-x-z & 2y \\ 2z & 2z & z-x-y \end{vmatrix} = 0$, this simplifies to a constraint on $x, y, z$.
Correct Answer: 0