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

Question

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

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

LohaHridoy

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2025-01-04 04:42

1 Answers

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AhmedNazir · Accepted Answer · 2022-09-04T04:29:57+06:00

Option 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.

Trace is the sum of all diagonal elements of a square matrix.

The determinant of a matrix = Product of Eigenvalues.

In a case where the trace is positive, and determinant is negative, then at least one eigenvalue of the matrix or an odd number of eigenvalues has to be negative because the product of eigenvalues of a given matrix is equal to the determinant of a given matrix.

Hence, to have the determinant negative, at least one Eigen value has to be negative but reverse may or may not be true.