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 + AT and B + BT are symmetric matrices.
• D. AB - BA is a symmetric matrix.

Hint:

The difference of two matrices AB - BA is not symmetric, even if A and B are symmetric matrices.

Answer 1

The Correct Option is D

A matrix A is symmetric if AT = A. Similarly, B is symmetric if BT = B. If A and B are symmetric matrices, then A + B will also be symmetric. 
This is because: (A + B)T = AT + BT = A + B Hence, A + B is a symmetric matrix. AB + BA is symmetric because: (AB + BA)T = BT AT + AT BT = BA + AB = AB + BA Thus, AB + BA is a symmetric matrix. A + AT and B + BT are symmetric matrices because: (A + AT)T = AT + A = A + AT and (B + BT)T = BT + B = B + BT Therefore, both A + AT and B + BT are symmetric matrices. AB - BA is generally not symmetric because: (AB - BA)T = BT AT - AT BT = BA - AB Thus, AB - BA is not equal to AB - BA, meaning it is not a symmetric matrix. The correct answer is (D), as AB - BA is not a symmetric matrix.

 Final Answer: (D) AB - BA is a symmetric matrix.