Question

The eigenvalues of the matrix 

are \( \lambda_1, \lambda_2, \lambda_3 \). The value of \( \lambda_1 \lambda_2 \lambda_3 ( \lambda_1 + \lambda_2 + \lambda_3 ) \) is: 
 

• A. 11
• B. 45
• C. 396
• D. 495

Hint:

When solving problems involving eigenvalues of matrices, the determinant of the matrix gives the product of the eigenvalues. The trace of the matrix gives the sum of the eigenvalues, which is useful for calculating various expressions involving eigenvalues.

Answer 1

The Correct Option is C

The given matrix is a \( 3 \times 3 \) matrix:

\[ A = \begin{bmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & -3 \end{bmatrix} \]

The determinant of the matrix can be used to calculate the product of the eigenvalues \( \lambda_1, \lambda_2, \lambda_3 \). After solving for the eigenvalues using the characteristic equation, we find:

\[ \lambda_1 = 5, \quad \lambda_2 = -3, \quad \lambda_3 = 1 \]

Next, we calculate the required expression:

\[ \lambda_1 \lambda_2 \lambda_3 ( \lambda_1 + \lambda_2 + \lambda_3 ) = 5 \times (-3) \times 1 \times (5 + (-3) + 1) \] \[ = (-15) \times 3 = -45 \]

Thus, the value is \( -45 \), and the correct answer is (B).