Question

Let \( A \in M_n({C}) \) be a normal matrix. Consider the following statements:
1. If all the eigenvalues of \( A \) are real, then \( A \) is Hermitian.
2. If all the eigenvalues of \( A \) have absolute value 1, then \( A \) is unitary.
Which one of the following is correct?

• A. Both I and II are TRUE
• B. I is TRUE and II is FALSE
• C. I is FALSE and II is TRUE
• D. Both I and II are FALSE

Hint:

For normal matrices, use the properties of eigenvalues to determine Hermitian or unitary nature.

Answer 1

The Correct Option is A

Step 1: Verifying the Hermitian property. If all eigenvalues of a normal matrix are real, then the matrix is Hermitian by definition. Step 2: Verifying the unitary property. If all eigenvalues of a normal matrix have absolute value 1, then the matrix is unitary by definition. Step 3: Conclusion. Both statements are true. The correct answer is \( {(1)} \).