Question

Which of the following matrices is Hermitian as well as unitary?

• A. \(\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\)
• B. \(\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}\)
• C. \(\begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix}\)
• D. \(\begin{pmatrix} 0 & 1+i \\ 1-i & 0 \end{pmatrix}\)

Answer 1

The Correct Option is A

1. Matrix: \( \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \)
   Check:
   - Hermitian: Yes (Transpose + conjugate gives the original matrix)
   - Unitary: Yes (Inverse equals conjugate transpose)
   Result: This matrix is Hermitian and Unitary.
2. Matrix: \( \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \)
   Check:
   - Hermitian: No (Transpose + conjugate does not give the original matrix)
   - Unitary: Not applicable
   Result: Not Hermitian and Unitary.
3. Matrix: \( \begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix} \)
   Check:
   - Hermitian: Yes (Transpose + conjugate gives the original matrix)
   - Unitary: No (Inverse does not equal conjugate transpose)
   Result: Not both Hermitian and Unitary.
4. Matrix: \( \begin{pmatrix} 0 & 1+i \\ 1-i & 0 \end{pmatrix} \)
   Check:
   - Hermitian: Yes (Transpose + conjugate gives the original matrix)
   - Unitary: No (Inverse does not equal conjugate transpose)
   Result: Not both Hermitian and Unitary.

Conclusion:

The matrix \( \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \) is Hermitian and Unitary.