Matrix Properties

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If A and B are symmetric matrices of the same order, then AB is symmetric if

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1. **Symmetric Constraint**: We require $(AB)^T = AB$.
2. **Transpose Property**: Note that $(AB)^T = B^T A^T$.
3. **Application**: Since $A$ and $B$ are symmetric, $A^T = A$ and $B^T = B$.
   - Substituting: $(AB)^T = B \, A$.
4. **Equivalence**: For $B \, A$ to equal $A \, B$, the matrices must **commute ($AB = BA$)**. If they don't commute, the product is not symmetric.

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