Question

The eigenvalues of

\[ \begin{pmatrix} 3 & i & 0 \\ -i & 3 & 0 \\ 0 & 0 & 6 \end{pmatrix} \]

are

• A. 2, 4 and 6
• B. 2i, 4i and 6
• C. 2i, 4 and 8
• D. 0, 4 and 8

Hint:

To find the eigenvalues of a matrix, solve the characteristic equation \( \det(A - \lambda I) = 0 \).

Answer 1

The Correct Option is A

Step 1: Understanding eigenvalues.
To find the eigenvalues of the matrix, we solve the characteristic equation:

\[ \det(A - \lambda I) = 0 \]

where \( A \) is the matrix, \( \lambda \) is the eigenvalue, and \( I \) is the identity matrix.
Step 2: Solving for eigenvalues.
For the given matrix:

\[ A = \begin{pmatrix} 3 & i & 0 \\ -i & 3 & 0 \\ 0 & 0 & 6 \end{pmatrix} \]

we find the characteristic equation by subtracting \( \lambda \) from the diagonal elements and computing the determinant. The resulting eigenvalues are \( 2i, 4, 8 \).
Step 3: Conclusion.
Thus, the eigenvalues of the matrix are \( 2i, 4, \) and \( 8 \), so the correct answer is (C).