Question

If A and B are symmetric matrices of the same order, which one of the following is not correct?

• A. \( A + B \) is a symmetric matrix.
• B. \( AB + BA \) is a symmetric matrix.
• C. \( A + A^T \) and \( B + B^T \) are symmetric matrices.
• D. \( AB - BA \) is a symmetric matrix.

Hint:

The sum of symmetric matrices is symmetric, but the difference is not necessarily symmetric due to non-commutative matrix multiplication.

Answer 1

The Correct Option is D

Step 1: Symmetric Matrix Definition. A matrix \( A \) is symmetric if \( A = A^T \). Both matrices \( A \) and \( B \) provided are symmetric.

Step 2: Option Evaluation. - (A) \( A + B \) is symmetric: True. The sum of two symmetric matrices yields a symmetric matrix. - (B) \( AB + BA \) is symmetric: True. The sum \( AB + BA \) is always symmetric. - (C) \( A + A^T \) and \( B + B^T \) are symmetric: True. Given \( A \) and \( B \) are symmetric, \( A + A^T \) and \( B + B^T \) are also symmetric. - (D) \( AB - BA \) is symmetric: False. \( AB - BA \) is typically not symmetric due to the non-commutative nature of matrix multiplication.

Step 3: Final Determination. Statement (D) is incorrect because \( AB - BA \) is generally not a symmetric matrix.