Question

Two eigenvalues of the following matrix are 3 and 6. The third eigenvalue is \[ \begin{bmatrix} -2 & -4 & 2 \\ -2 & 1 & 2 \\ 4 & 2 & 5 \end{bmatrix} \]

• A. -5
• B. -1
• C. 1
• D. 4

Hint:

- The sum of the eigenvalues of a matrix equals the trace of the matrix.
- For a \(3 \times 3\) matrix, the trace is the sum of the diagonal elements.

Answer 1

The Correct Option is A

Given that the matrix is a \(3 \times 3\) matrix, the sum of its eigenvalues is equal to the trace of the matrix. The trace is the sum of the diagonal elements of the matrix. The matrix is: \[ A = \begin{bmatrix} -2 & -4 & 2 \\ -2 & 1 & 2 \\ 4 & 2 & 5 \end{bmatrix} \] The trace of the matrix is: \[ \text{Tr}(A) = (-2) + 1 + 5 = 4 \] The sum of the eigenvalues is the trace of the matrix, which is 4. We are given that two of the eigenvalues are 3 and 6. Let the third eigenvalue be \( \lambda_3 \). Therefore: \[ 3 + 6 + \lambda_3 = 4 \] Solving for \( \lambda_3 \): \[ \lambda_3 = 4 - 9 = -5 \] Therefore, the third eigenvalue is: \[ \boxed{-5} \]