Question

If $\omega$ is an imaginary cube root of unity,then the value of
$\left|\begin{array}{ccc}a& b{\omega }^2& a\omega \\ b\omega & c& b{\omega }^2\\ c{\omega }^2& a\omega & c\end{array}\right|$,is ?

• A. $a^3+b^3+c^3$
• B. $a^{2}b-b^{2}c$
• C. 0
• D. $a^{3}b+b^3+3abc$

Answer 1

Answer: (C)

0

$△=\left|\begin{array}{ccc}a& b{\omega }^2& a\omega \\ b\omega & c& b{\omega }^2\\ c{\omega }^2& a\omega & c\end{array}\right|1+\omega +{\omega }^2=0;{\omega }^3=1;{\omega }^4=\omega$

$\Rightarrow a(c^2-ab{\omega }^3)-b{\omega }^2(bc\omega -bc{\omega }^4)+a\omega (ab{\omega }^2-c^2{\omega }^2)$

$\Rightarrow a(c^2-ab{\omega }^3)-b{\omega }^2(bc\omega -bc\omega )+a(ab-c^2)$

$=a(c^2-ab)-a(c^2-ab)=0$

$\therefore △=0$

Option C