Which one of the following statements is TRUE about every ݊\(n \times n\)…
2014
Which one of the following statements is TRUE about every ݊\(n \times n\) matrix with only real eigenvalues?
- A.
If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative
- B.
If the trace of the matrix is positive, all its eigenvalues are positive.
- C.
If the determinant of the matrix is positive, all its eigenvalues are positive.
- D.
If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.
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Correct answer: A
Key insight: for a matrix with only real eigenvalues, the determinant equals the product of the eigenvalues and the trace equals their sum.
Why the given true statement holds: If the determinant is negative then the product of the eigenvalues is negative, so an odd number of eigenvalues are negative. Therefore at least one eigenvalue is negative. The stated requirement that the trace is positive does not change this implication (it is unnecessary but does not make the statement false).
Counterexamples to the other statements:
A positive trace does not force all eigenvalues to be positive. Example: eigenvalues 5 and -1 have trace 4 (positive) but one eigenvalue is negative.
A positive determinant does not force all eigenvalues to be positive. Example: eigenvalues -2 and -3 have determinant 6 (positive) while both eigenvalues are negative.
A positive product of trace and determinant does not guarantee all eigenvalues are positive. Example: eigenvalues 4, -1, -1 give trace 2 (positive) and determinant 4 (positive), so the product is positive, yet two eigenvalues are negative.
Conclusion: The statement that combines a positive trace with a negative determinant and concludes at least one eigenvalue is negative is correct because a negative determinant forces an odd number of negative eigenvalues (hence at least one).