Let A be a 3 × 3 matrix with real entries. Which of the following assertions is FALSE?

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CSIR UGC (NET) Mathematical Science: Held On (7 June 2023)
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  1. A must have a real eigenvalue.
  2. If the determinant of A is 0 , then 0 is an eigenvalue of A.
  3. If the determinant of A is negative and 3 is an eigenvalue of A, then A must have three real eigenvalues.
  4. If the determinant of A is positive and 3 is an eigenvalue of A, then A must have three real eigenvalues.

Answer (Detailed Solution Below)

Option 4 : If the determinant of A is positive and 3 is an eigenvalue of A, then A must have three real eigenvalues.
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Concept:

Odd degree polynomial must have at least one real root

Explanation:

A is a a 3 × 3 matrix with real entries.

So characteristic polynomial of A will be of degree 3.

(1): Since we know that, odd degree polynomial must have at least one real root so A must have a real eigenvalue.

(1) is true

(2): As we know that determinant of a matrix is equal to the product of eigenvalues. So if the determinant of A is 0, then 0 is an eigenvalue of A.

(2) is true

(3): The determinant of A is negative and 3 is an eigenvalue of A.

If possible let the other two eigenvalues of A are not real and they are α + iβ, α - iβ

So determinant = 3(α + iβ)(α - iβ) = 3(α2 + β2) > 0 for all α, β which is a contradiction.

So A must have three real eigenvalues.

(3) is true and (4) is false statement    

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