Let A and B be real symmetric matrices of the same size. Which one of the following options is correct?

The correct option is:
$AB=BA$

Step-by-Step Explanation:

Let us analyze the definition of a symmetric matrix. A real matrix A is defined to be symmetric if it is equal to its transpose. Mathematically, this is expressed as:
$A^T=A$
Similarly, since B is also a real symmetric matrix of the same size, we have:
$B^T=B$

Now, let us examine the properties of matrix multiplication and transposes. For any two matrices A and B for which the product is defined, the transpose of their product satisfies the reversal law:
${\left(AB\right)}^T=B^T{A}^T$
Since A and B are symmetric, we can substitute $A^T=A$ and $B^T=B$ into the reversal formula:
${\left(AB\right)}^T=BA$

For the product of two symmetric matrices $A$ and $B$ to also be symmetric, it must satisfy:
${\left(AB\right)}^T=AB$
Equating the two expressions for the transpose of the product gives the necessary and sufficient condition:
$AB=BA$
Therefore, the matrices commute. This confirms that the given option is correct under the standard properties of symmetric matrix multiplication.