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Which one of the following statements is TRUE about every $n \times n$ matrix with only real eigenvalues?

1. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.
2. If the trace of the matrix is positive, all its eigenvalues are positive.
3. If the determinant of the matrix is positive, all its eigenvalues are positive.
4. If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.

Correct answer: If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.

Solution

For a matrix with all real eigenvalues, the determinant equals the product of eigenvalues. If the determinant is negative, the product of real numbers is negative, so at least one eigenvalue must be negative. The other statements are not necessarily true because positive trace or determinant alone does not force all eigenvalues to be positive.

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