Question

The eigenvalues of \(\begin{bmatrix} 0 & -i \\[0.3em]i & 0 \end{bmatrix}\)are ____ .

• A. \(i, i\)
• B. \(i, -i\)
• C. \(-1, -1\)
• D. \(-1, 1\)

Hint:

In matrix algebra, eigenvalues are calculated by solving the characteristic equation for the determinant of \( \mathbf{A} - \lambda \mathbf{I} \).

Answer 1

The Correct Option is D

To find the eigenvalues of the matrix \( \begin{bmatrix} 0 & -i
i & 0 \end{bmatrix} \), we must solve the characteristic equation: \[ \text{det} \left( \begin{bmatrix} 0 & -i
i & 0 \end{bmatrix} - \lambda I \right) = 0 \] which simplifies to: \[ \begin{vmatrix} -\lambda & -i
i & -\lambda \end{vmatrix} = 0 \] This results in the equation: \[ \lambda^2 + 1 = 0 \] Solving for \( \lambda \), we get: \[ \lambda = \pm i \] Thus, the eigenvalues are **\(-1, 1\)**.