Question

If $w$ is a complex cube root of unity, then value of $\Delta =\left|\begin{array}{ccc}{a}_1+b_{1}w& a_1{w}^2+b_1& c_1+b_1\overline{w}\\ a_2+b_{2}w& a_2{w}^2+b_2& c_2+b_2\overline{w}\\ a_3+b_{3}w& a_3{w}^2+b_3& c_3+b_3\overline{w}\end{array}\right|$

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

Answer 1

Answer: (A)

$0$

$△=\left|\begin{array}{ccc}{a}_1+b_1\omega & a_1{\omega }^2+b_1& c_1+b_1\overline{\omega }\\ a_2+b_2\omega & a_2{\omega }^2+b_2& c_2+b_2\overline{\omega }\\ a_3+b_3\omega & a_3{\omega }^2+b_3& c_3+b_3\overline{\omega }\end{array}\right|$

$△={\omega }^2\times \frac{△}{{\omega }^2}={\omega }^2\times \frac{1}{{\omega }^2}\left|\begin{array}{ccc}{a}_1+b_1\omega & a_1{\omega }^2+b_1& c_1+b_1\overline{\omega }\\ a_2+b_2\omega & a_2{\omega }^2+b_2& c_2+b_2\overline{\omega }\\ a_3+b_3\omega & a_3{\omega }^2+b_3& c_3+b_3\overline{\omega }\end{array}\right|$

${\omega }^2$ is multiplied to$C_1$

${\omega }^3=1$

$\Rightarrow \frac{1}{{\omega }^2}\left|\begin{array}{ccc}{a}_1{\omega }^2+b_1& a_1{\omega }^2+b_1& c_1+b_1\overline{\omega }\\ a_2{\omega }^2+b_2& a_2{\omega }^2+b_2& c_2+b_2\overline{\omega }\\ a_3{\omega }^2+b_3& a_3{\omega }^2+b_3& c_3+b_3\overline{\omega }\end{array}\right|$

$C_1$ and $C_2$ are same

$\Rightarrow △=0$

Option A is correct