Determine Whether ABA^T is Symmetric or Skew-Symmetric

Given:

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

Step 1: Take Transpose

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

Step 2: Use Properties

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

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

Conclusion:

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

Hence, ABA^T is symmetric.