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

Step 1: Formula / Definition}
\[ |z_1|^2 + |z_2|^2 = 1 + 1 = 2 \quad (\text{unimodular}) \]
Step 2: Calculation / Simplification}
Let \(U = \begin{bmatrix} z_1 & z_2\\ -\bar{z}_2 & \bar{z}_1 \end{bmatrix}\). Then \(U^{-1} = \frac{1}{2}\begin{bmatrix} \bar{z}_1 & -z_2 \\ \bar{z}_2 & z_1 \end{bmatrix}\)
The given product is \(U^{-1} (U^{-1})^* = U^{-1} (U^*)^{-1} = (U^* U)^{-1}\)
\(U^* U = \begin{bmatrix} |z_1|^2+|z_2|^2 & 0 \\ 0 & |z_1|^2+|z_2|^2 \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}\)
\((U^* U)^{-1} = \begin{bmatrix} 1/2 & 0 \\ 0 & 1/2 \end{bmatrix}\)
Step 3: Final Answer
\[ \begin{bmatrix} 1/2 & 0 \\ 0 & 1/2 \end{bmatrix} \]