Question
Download Solution PDFLet A be a 3 × 3 matrix with real entries. Which of the following assertions is FALSE?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
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
Last updated on Jun 5, 2025
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