Question

Two of the eigenvalues of a 3x3 matrix are -1 and 2. The determinant of the matrix is 4. Then the third eigenvalue is

• A. The determinant of the matrix is
• B. Then the third eigenvalue is}
• C. -1
• D. -2

Hint:

The determinant of a matrix is the product of its eigenvalues. If you know the determinant and some eigenvalues, you can solve for the remaining eigenvalues.

Answer 1

The Correct Option is B

The determinant of a matrix is the product of its eigenvalues. For a 3x3 matrix with eigenvalues \( \lambda_1 \), \( \lambda_2 \), and \( \lambda_3 \), we have: \[ \text{det}(A) = \lambda_1 \cdot \lambda_2 \cdot \lambda_3 \] Given that \( \lambda_1 = -1 \), \( \lambda_2 = 2 \), and \( \text{det}(A) = 4 \), we substitute into the equation: \[ 4 = (-1) \cdot 2 \cdot \lambda_3 \] Solving for \( \lambda_3 \): \[ \lambda_3 = \frac{4}{-1 \cdot 2} = \frac{4}{-2} = -2 \]
Therefore, the third eigenvalue is -2.