Question

The determinant of a \( 3 \times 3 \) matrix A is 30. If 2 and 3 are two Eigenvalues of A, then the third Eigenvalue of A (in integer) is ________________.

Hint:

For a square matrix, the determinant is the product of its Eigenvalues.

Answer 1

Correct Answer: 5

The determinant of a matrix is the product of its Eigenvalues. Given that two Eigenvalues of \( A \) are 2 and 3, we can use the following relation for the determinant of the matrix: \[ \text{det}(A) = \lambda_1 \times \lambda_2 \times \lambda_3. \] Substituting the given values: \[ 30 = 2 \times 3 \times \lambda_3. \] Solving for \( \lambda_3 \): \[ \lambda_3 = \frac{30}{6} = 5. \] Final Answer: \[ \boxed{5}. \]