Complex Numbers Question - None

Question

Let w = e^iπ/3, where i = √−1 and a, b, c, x, y and z be non-zero complex numbers such that

a + b + c = x

a + bω + cω² = y

a + bω² + cω = z

The value of $\frac{|x|^2 + |y|^2 + |z|^2}{|a|^2 + |b|^2 + |c|^2}$ is

Accepted Answer

Step 1: The system of equations can be written in matrix form as:

$$\begin{pmatrix} 1 & 1 & 1 \ 1 & \omega & \omega^2 \ 1 & \omega^2 & \omega \end{pmatrix} \begin{pmatrix} a \ b \ c \end{pmatrix} = \begin{pmatrix} x \ y \ z \end{pmatrix}$$

Step 2: The matrix is a DFT-type matrix with ω = e^(2πi/3), and its rows are orthogonal.

Step 3: By Parseval's theorem (or orthogonality of DFT), $|x|^2 + |y|^2 + |z|^2 = 3(|a|^2 + |b|^2 + |c|^2)$.

∴ The value is 3.

Question:

Step-by-Step Solution

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