Question

Let A be a 3 \(\times\) 3 matrix such that the characteristic polynomial of A is f(x) \(=\) x\(^3\) \(-\) 4x\(^2\) \(-\) 11x \(+\) 30. If one of the eigenvalues of A is \(-\)3, then sum of the other two eigenvalues is

• A. \(-1\)
• B. \(-10\)
• C. 4
• D. 7

Hint:

For any characteristic equation \( x^3 - c_1 x^2 + c_2 x - c_3 = 0 \):
- Sum of roots (eigenvalues) \( = c_1 \) (coefficient of \( x^2 \) with sign changed).
- Product of roots (determinant) \( = c_3 \) (constant term with sign changed).
This property saves valuable time by avoiding the need to fully factor the polynomial.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question asks for the sum of two unknown eigenvalues of a \( 3 \times 3 \) matrix, given its characteristic polynomial and one of its eigenvalues.

Step 2: Key Formula or Approach:
Let \( \lambda_1, \lambda_2, \lambda_3 \) be the three eigenvalues of matrix \( A \).
The characteristic equation of a \( 3 \times 3 \) matrix \( A \) is represented as:
\[ x^3 - \text{tr}(A) \cdot x^2 + M \cdot x - |A| = 0 \]
where:
\( \text{tr}(A) \) is the trace of the matrix (sum of the diagonal elements).
\( |A| \) is the determinant of the matrix.
A key theorem in linear algebra states that the sum of the eigenvalues is equal to the trace of the matrix:
\[ \sum_{i=1}^3 \lambda_i = \lambda_1 + \lambda_2 + \lambda_3 = \text{tr}(A) \]

Step 3: Detailed Explanation:

• Identify the Trace from the Polynomial:
- The given characteristic polynomial is:
\[ f(x) = x^3 - 4x^2 - 11x + 30 \]
- Comparing this to the standard form \( x^3 - \text{tr}(A) \cdot x^2 + M \cdot x - |A| \), we find:
\[ \text{tr}(A) = 4 \]
- Therefore, the sum of all three eigenvalues must be:
\[ \lambda_1 + \lambda_2 + \lambda_3 = 4 \]

• Calculate the Sum of the Other Two Eigenvalues:
- We are given that one of the eigenvalues is \( -3 \). Let \( \lambda_1 = -3 \).
- Substitute this into the sum equation:
\[ -3 + \lambda_2 + \lambda_3 = 4 \]
\[ \lambda_2 + \lambda_3 = 4 + 3 = 7 \]
- Thus, the sum of the other two eigenvalues is exactly 7.


Step 4: Final Answer:
The sum of the other two eigenvalues is 7.
Therefore, the correct choice is option (D).