Question

Let $x=\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}$ and $\Delta =\left|\begin{array}{ccc}1& x& x^2\\ x^2& 1& x\\ 1& x^2& 1\end{array}\right|$ then numerical value of $\Delta$ is

• A. 0
• B. -1
• C. 8
• D. -4

Answer 1

Answer: (C)

Use $C_3\to C_3-x{C}_2$ and $C_2\to C_2-x{C}_1$.
$\Delta =\left|\begin{array}{ccc}1& 0& 0\\ x^2& 1-x^3& 0\\ 1& x^2-x& 1-x^3<br/>\end{array}\right|$
$={\left(1-x^3\right)}^2=[1-\cos \pi -i\sin \pi {]}^2=8$
[De Moivre's Theorem]