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: Understanding symmetric matrices.
A matrix is symmetric if \( A = A^T \), i.e., it is equal to its transpose. In this question, both \( A \) and \( B \) are symmetric matrices.

Step 2: Analysis of options.
- (A) \( A + B \) is a symmetric matrix: This is correct. The sum of two symmetric matrices is always symmetric.
- (B) \( AB + BA \) is a symmetric matrix: This is correct. The sum of \( AB \) and \( BA \) is symmetric.
- (C) \( A + A^T \) and \( B + B^T \) are symmetric matrices: This is correct. Since \( A \) and \( B \) are symmetric, \( A + A^T \) and \( B + B^T \) are also symmetric.
- (D) \( AB - BA \) is a symmetric matrix: This is incorrect. The difference \( AB - BA \) is generally not symmetric, as matrix multiplication is not commutative.

Step 3: Conclusion.
The incorrect statement is (D), as \( AB - BA \) is generally not a symmetric matrix.