Question

For two unimodular complex numbers \(z_1\) and \(z_2\), \(\begin{bmatrix} z_1 & z_2 \\ -\bar{z}_2 & \bar{z}_1 \end{bmatrix}^{-1} \begin{bmatrix} \bar{z}_1 & -z_2 \\ \bar{z}_2 & z_1 \end{bmatrix}^{-1}\) is equal to

• A. \(\begin{bmatrix} z_1 & z_2 \\ z_1 & z_2 \end{bmatrix}\)
• B. \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)
• C. \(\begin{bmatrix} 1/2 & 0 \\ 0 & 1/2 \end{bmatrix}\)
• D. None of these

Hint:

Use \(U^{-1} = \frac{1}{\det U} \text{adj}(U)\) and properties of unitary matrices.

Answer 1

The Correct Option is C