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Home » Mark the tick against the correct answer in the following: If is a complex root of unity then A. 1 B. C. 0 D. none of these
Mark the tick against the correct answer in the following: If \omega is a complex root of unity then \left|\begin{array}{ccc}1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega\end{array}\right|=?
A. 1
B. -1
C. 0
D. none of these
CBSE | Class 12 | Determinants | Maths | RS Aggarwal
Mark the tick against the correct answer in the following: If \omega is a complex root of unity then \left|\begin{array}{ccc}1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega\end{array}\right|=?
A. 1
B. -1
C. 0
D. none of these

Solution:

Option(C)
To find: Value of \left|\begin{array}{ccc}1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega\end{array}\right|
We have, \left|\begin{array}{ccc}1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega\end{array}\right|
On expanding the above along 1^{\text {st }} column
\begin{array}{l} \Rightarrow 1\left|\begin{array}{cc} \omega^{2} & 1 \\ 1 & \omega \end{array}\right|-\omega\left|\begin{array}{cc} \omega & \omega^{2} \\ 1 & \omega \end{array}\right|+\omega^{2}\left|\begin{array}{cc} \omega & \omega^{2} \\ \omega^{2} & 1 \end{array}\right| \\ \Rightarrow\left[1\left\{\left(\omega^{2}\right)(\omega)-(1)(1)\right\}\right]-\left[\omega\left\{(\omega)(\omega)-\left(\omega^{2}\right)(1)\right\}\right]+\left[\omega^{2}\left\{(\omega)(1)-\left(\omega^{2}\right)\left(\omega^{2}\right)\right\}\right] \\ \Rightarrow\left[1\left\{\omega^{3}-1\right\}\right]-\left[\omega\left\{\omega^{2}-\omega^{2}\right\}\right]+\left[\omega^{2}\left\{\omega-\omega^{4}\right\}\right] \ldots \text { (i) } \end{array}
As \omega^{3}=1
\begin{array}{l} \Rightarrow \omega^{3} \cdot \omega=1 \cdot \omega \\ \Rightarrow \omega^{4}=\omega \end{array}Using the above obtained value of \omega^{4} in eqn. (i)
\begin{array}{l} \Rightarrow\left[1\left\{\omega^{3}-1\right\}\right]-\left[\omega\left\{\omega^{2}-\omega^{2}\right\}\right]+\left[\omega^{2}\{\omega-\omega\}\right] \\ \Rightarrow 1\left\{\omega^{3}-1\right\} \\ \Rightarrow \omega^{3}-1 \\ \Rightarrow 1-1=0 \end{array}

i

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