Question

If A is an n x n real matrix, then:

• A. The eigenvalues of A are real or complex conjugates in pairs
• B. The eigenvalues of A are all equal
• C. The eigenvalues of A cannot be complex numbers
• D. The eigenvalues of A are all positive real numbers

Hint:

Exam Tip:
For any real matrix:

• The characteristic polynomial has real coefficients.
• Complex eigenvalues occur in conjugate pairs.
• The trace equals the sum of eigenvalues.
• The determinant equals the product of eigenvalues.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question tests knowledge of eigenvalues of a real matrix. The characteristic polynomial of a real matrix has real coefficients.

Step 2: Key Properties of Eigenvalues:
For a real matrix \(A\) of size \(n \times n\):
• The characteristic polynomial \(p(\lambda) = \det(A - \lambda I)\) has real coefficients.
• If \(\lambda\) is a complex eigenvalue, then its complex conjugate \(\bar{\lambda}\) is also an eigenvalue.
• The eigenvalues can be real or occur in complex conjugate pairs.
• There is no condition that all eigenvalues must be equal, positive, or real.

Step 3: Analyzing the Options:

• (A) The eigenvalues of A are real or complex conjugates in pairs:
This is true. For a real matrix, complex eigenvalues always come in conjugate pairs.
• (B) The eigenvalues of A are all equal:
This is false. Only special matrices like scalar multiples of the identity have all equal eigenvalues.
• (C) The eigenvalues of A cannot be complex numbers:
This is false. A real matrix can have complex eigenvalues (e.g., rotation matrices).
• (D) The eigenvalues of A are all positive real numbers:
This is false. Eigenvalues can be negative, zero, or complex.

Step 4: Final Answer:
The eigenvalues of a real matrix are either real or occur in complex conjugate pairs. Therefore, option (A) is correct.