Question

The imaginary parts of the eigenvalues of the matrix 

are 
 

• A. 0, 0, 0
• B. 2, -2, 0
• C. 1, -1, 0
• D. 3, -3, 0

Hint:

When calculating eigenvalues, solving the characteristic equation \( \det(P - \lambda I) = 0 \) will give you the eigenvalues, and you can determine if there are any imaginary components by analyzing the roots.

Answer 1

The Correct Option is A

Step 1: Finding the characteristic equation. 
To find the eigenvalues of a matrix \( P \), we need to solve the characteristic equation, which is given by: \[ \det(P - \lambda I) = 0 \] where \( I \) is the identity matrix and \( \lambda \) is the eigenvalue. In this case, the matrix \( P - \lambda I \) becomes: 

Step 2: Solving the determinant. 
Now, calculate the determinant of this matrix: 

The determinant simplifies and leads to the characteristic equation. After solving, we find that all the eigenvalues are real numbers with no imaginary part. Thus, the imaginary parts of the eigenvalues are all \( 0 \). Therefore, the correct answer is (A).