If A and B are symmetric matrices of the same order, then show that AB is symmetric if and only if A and B commute, that is `A B = B A`.

Explanation

To show that if \( A \) and \( B \) are symmetric matrices of the same order, then \( AB \) is symmetric if and only if \( A \) and \( B \) commute (i.e., \( AB = BA \)), we will prove both directions of the statement.

### Step 1: Prove that if \( AB \) is symmetric, then \( AB = BA \)

1. **Assumption**: Assume \( AB \) is symmetric. By the definition of a symmetric matrix, we have:
\[ (AB)^T = AB \]
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