Question

If $\omega $ is a complex cube root of unity, then value of
$\Delta=\begin{vmatrix}a_{1}+b_{1} \omega&a_{1} \omega^{2}+b_{1}&c_{1}+b_{1} \bar{\omega}\\ a_{2}+b_{2}\omega&a_{2}\omega^{2}+b_{2}&c_{2}+b_{2}\bar{\omega}\\ a_{3}+b_{3}\omega&a_{3}\omega^{2}+b_{3}&c_{3}+b_{3}\bar{\omega}\end{vmatrix}$ is

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

Answer 1

Answer: (A)

$\Delta=\begin{vmatrix}a_{1}+b_{1} \omega&a_{1} \omega^{2}+b_{1}&c_{1}+b_{1} \bar{\omega}\\ a_{2}+b_{2}\omega&a_{2}\omega^{2}+b_{2}&c_{2}+b_{2}\bar{\omega}\\ a_{3}+b_{3}\omega&a_{3}\omega^{2}+b_{3}&c_{3}+b_{3}\bar{\omega}\end{vmatrix}$
Using $C_{2} \rightarrow \omega C_{2}$
We have $\Delta=\frac{1}{\omega}\begin{vmatrix}a_{1}+b_{1} \omega&a_{1} \omega^{3}+b_{1} \omega&c_{1}+b_{1} \bar{\omega}\\ a_{2}+b_{2}\omega&a_{2}\omega^{3}+b_{2} \omega&c_{2}+b_{2}\bar{\omega}\\ a_{3}+b_{3}\omega&a_{3}\omega^{3}+b_{3}\omega&c_{3}+b_{3}\bar{\omega}\end{vmatrix}$
$=\frac{1}{\omega}\begin{vmatrix}a_{1}+b_{1} \omega&a_{1} +b_{1} \omega&c_{1}+b_{1} \bar{\omega}\\ a_{2}+b_{2}\omega&a_{2}+b_{2} \omega&c_{2}+b_{2}\bar{\omega}\\ a_{3}+b_{3}\omega&a_{3}+b_{3}\omega&c_{3}+b_{3}\bar{\omega}\end{vmatrix}=0$