Condition for AB to be Symmetric
Given:
\[ A \text{ and } B \text{ are symmetric matrices} \Rightarrow A^T = A,\; B^T = B \]
Step 1: Take Transpose of AB
\[ (AB)^T = B^T A^T \]
Step 2: Use Symmetry
\[ (AB)^T = BA \]
Step 3: Condition for Symmetry
\[ AB \text{ is symmetric if } (AB)^T = AB \]
\[ \Rightarrow BA = AB \]
Final Answer:
\[ \boxed{AB = BA} \]
Thus, AB is symmetric only when A and B commute.