Question

If $a=\cos \frac{4\pi }{3}+i\sin \frac{4\pi }{3}$ then $\left|\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& a^2& a\end{array}\right|$ =

• A. purely real
• B. purely imaginary
• C. complex number
• D. rational

Answer 1

The correct option is B purely imaginary

$a=cos\frac{4\pi }{3}+jsin\frac{4\pi }{3}=w$
w is a cube root of unity
So, $A=\left|\begin{array}{ccc}1& 1& 1\\ 1& a^2& a^2\\ 1& a^2& a\end{array}\right|=1(a^2-a^4)-1(a-a^2)-1(a-a^2)+1(a^2-a)$
$=2a^2-a^4-a+a^2-a$
$A=3a^2-a^4-2a$
So, $w^3=1and1+w+w^2=0$
So, $A=3a^2-3a=3a^2-3(-a^2-1)$
$3(a^2+1)$
$3(-a)$
$(\because 1+a+a^2)=0$