Question

If $x$ is a complex root of the equation
$\left|\begin{matrix}1&x&x\\ x&1&x\\ x&x&1\end{matrix}\right|+\left|\begin{matrix}1-x&1&1\\ 1&1-x&1\\ 1&1&1-x\end{matrix}\right|=0$ , then $x^{2007}+ x^{-2007}=$

• A. $1$
• B. $-1$
• C. $-2$
• D. $2$

Answer 1

Answer: (C)

Expanding the two determinants, we get

$\left(1-3x^{2}+2x^{3}\right)+\left(3x^{2}-x^{3}\right)=0$

$\Rightarrow \quad x^{3}+1=0 \quad\Rightarrow x=-\omega, -\omega^{2}, -1$

$x^{2007}+x^{-2007}=-1-1=-2 $ .