(i) Find all the values of the complex number c such that det (c^2 A)= det. A, where A is a thi

step 1 If we want to show that there exists two unique W, such that W squared is equal to Z for all com

(i) Find all the values of the complex number c such that...

(i) Find all the values of the complex number $c$ such that det $\left(\mathrm{c}^{2} \mathrm{~A}\right)=$ det. $\mathrm{A}$, where $\mathrm{A}$ is a third order non-singular matrix. (ii) If $\mathrm{A}$ is an nth order non-singular matrix then show that all the complex numbers satisfying the above condition lie on a unit circle.

(i) Find all the values of the complex number $c$ such that det $\left(\mathrm{c}^{2} \mathrm{~A}\right)=$ det. $\mathrm{A}$, where $\mathrm{A}$ is a third order non-singular matrix.
(ii) If $\mathrm{A}$ is an nth order non-singular matrix then show that all the complex numbers satisfying the above condition lie on a unit circle.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Unit Circle in the Complex Plane

The unit circle consists of all complex numbers with an absolute value of 1. This concept is key in complex number theory since any complex number on the unit circle represents a pure rotation, a property that is often used to characterize the solutions of equations involving powers of complex numbers.

Determinant Scaling Property

This property states that when a square matrix of order n is multiplied by a scalar, the determinant of the new matrix becomes the scalar raised to the power n times the original determinant. It is a fundamental concept in linear algebra used to understand how scaling a matrix affects its volume-scaling factor and invertibility.

Non-Singular (Invertible) Matrix

A non-singular matrix is one that has a non-zero determinant, which guarantees the existence of its inverse. This concept is crucial for many matrix operations and ensures that certain equations, like those involving determinant scaling, have meaningful and non-trivial solutions.

Complex Roots of Unity

These are complex numbers that satisfy the equation z^m = 1 for some positive integer m. They have a multiplicative order and lie on the unit circle, making them important in problems where the solutions to equations are required to have a modulus of one, thus representing rotational symmetry in the complex plane.

Transcript

Please give Ace some feedback