Let omega be the complex number cos frac{2 pi | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given $\omega = e^{i 2 \pi / 3}$,we know that $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$.Let the determinant be $\Delta = \left|\begin{array}{

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(A) Given $\omega = e^{i 2 \pi / 3}$,we know that $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$.Let the determinant be $\Delta = \left|\begin{array}{ccc} z+1 & \omega & \omega^2 \ \omega & z+\omega^2 & 1 \ \omega^2 & 1 & z+\omega \end{array}ight|$.Applying the operation $C_1 	o C_1 + C_2 + C_3$,we get:$\Delta = \left|\begin{array}{ccc} z+1+\omega+\omega^2 & \omega & \omega^2 \ z+\omega+\omega^2+1 & z+\omega^2 & 1 \ z+\omega^2+1+\omega & 1 & z+\omega \end{array}ight| = \left|\begin{array}{ccc} z & \omega & \omega^2 \ z & z+\omega^2 & 1 \ z & 1 & z+\omega \end{array}ight|$.Factoring out $z$ from the first column:$\Delta = z \left|\begin{array}{ccc} 1 & \omega & \omega^2 \ 1 & z+\omega^2 & 1 \ 1 & 1 & z+\omega \end{array}ight|$.Applying $R_2 	o R_2 - R_1$ and $R_3 	o R_3 - R_1$:$\Delta = z \left|\begin{array}{ccc} 1 & \omega & \omega^2 \ 0 & z+\omega^2-\omega & 1-\omega^2 \ 0 & 1-\omega & z+\omega-\omega^2 \end{array}ight| = z[(z+\omega^2-\omega)(z+\omega-\omega^2) - (1-\omega^2)(1-\omega)]$.Expanding this,we find $\Delta = z(z^2) = z^3 = 0$.Thus,$z = 0$ is the only solution. The number of distinct complex numbers is $1$.

Let $\omega$ be the complex number $\cos \frac{2 \pi}{3} + i \sin \frac{2 \pi}{3}$. Then the number of distinct complex numbers $z$ satisfying $\left|\begin{array}{ccc} z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{array}\right| = 0$ is equal to

Similar Questions

The number of cubic polynomials $P(x)$ satisfying $P(1)=2, P(2)=4, P(3)=6, P(4)=8$ is

If $M$ and $N$ are square matrices of order $3$,then which one of the following statements is not true?

If $B$ is a $3 \times 3$ matrix such that $B^2 = 0$,then $\det[(I + B)^{50} - 50B]$ is equal to

If $A, B, C$ are the angles of a triangle,then the value of the determinant $\left| \begin{array}{ccc} \sin 2A & \sin C & \sin B \\ \sin C & \sin 2B & \sin A \\ \sin B & \sin A & \sin 2C \end{array} \right|$ is

The value of $(1+\Delta)(1-\nabla)$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module