Determine Whether ABA^T is Symmetric or Skew-Symmetric

Given:

\[ B \text{ is skew-symmetric } \Rightarrow B^T = -B \]

Step 1: Take Transpose

\[ (ABA^T)^T = (A^T)^T B^T A^T \]

Step 2: Use Properties

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

\[ (ABA^T)^T = A(-B)A^T = -ABA^T \]

Conclusion:

\[ (ABA^T)^T = -ABA^T \]

Hence, ABA^T is skew-symmetric.