Question

If $\omega$ is a complex cube root of unity, then value of
$\Delta =\left|\begin{array}{ccc}{a}_1+b_1\omega & a_1{\omega }^2+b_1& c_1+b_1\bar{\omega }\\ a_2+b_2\omega & a_2{\omega }^2+b_2& c_2+b_2\bar{\omega }\\ a_3+b_3\omega & a_3{\omega }^2+b_3& c_3+b_3\bar{\omega }\end{array}\right|$ is

• A. 0
• B. -1
• C. 2
• D. None of these

Answer 1

Answer: (A)

$\Delta =\left|\begin{array}{ccc}{a}_1+b_1\omega & a_1{\omega }^2+b_1& c_1+b_1\bar{\omega }\\ a_2+b_2\omega & a_2{\omega }^2+b_2& c_2+b_2\bar{\omega }\\ a_3+b_3\omega & a_3{\omega }^2+b_3& c_3+b_3\bar{\omega }\end{array}\right|$
Using $C_2\to \omega C_2$
We have $\Delta =\frac{1}{\omega }\left|\begin{array}{ccc}{a}_1+b_1\omega & a_1{\omega }^3+b_1\omega & c_1+b_1\bar{\omega }\\ a_2+b_2\omega & a_2{\omega }^3+b_2\omega & c_2+b_2\bar{\omega }\\ a_3+b_3\omega & a_3{\omega }^3+b_3\omega & c_3+b_3\bar{\omega }\end{array}\right|$
$=\frac{1}{\omega }\left|\begin{array}{ccc}{a}_1+b_1\omega & a_1+b_1\omega & c_1+b_1\bar{\omega }\\ a_2+b_2\omega & a_2+b_2\omega & c_2+b_2\bar{\omega }\\ a_3+b_3\omega & a_3+b_3\omega & c_3+b_3\bar{\omega }\end{array}\right|=0$