Determine Whether AB − BA is Symmetric or Skew-Symmetric

Given:

\[ A^T = A,\quad B^T = B \]

Step 1: Take Transpose

\[ (AB – BA)^T = (AB)^T – (BA)^T \]

Step 2: Use Property

\[ (AB)^T = B^T A^T,\quad (BA)^T = A^T B^T \]

\[ (AB – BA)^T = BA – AB \]

Step 3: Rearrange

\[ (AB – BA)^T = -(AB – BA) \]

Conclusion:

\[ (AB – BA)^T = -(AB – BA) \]

Hence, AB − BA is skew-symmetric.