Given:

• $A$ and $B$ are symmetric matrices of order $3$.
• So, $A^T=A,\qquad B^T=B.$

Statement–1: $A(BA)$ and $(AB)A$ are symmetric matrices

First note that matrix multiplication is associative, so $A(BA)=(AB)A=ABA.$

Now check symmetry of $ABA$: $(ABA)^T = A^T B^T A^T.$ Since $A$ and $B$ are symmetric, $(ABA)^T = ABA.$ Hence $ABA$ is symmetric.

Therefore both $A(BA)$ and $(AB)A$ are symmetric matrices.

So, Statement–1 is true.

Statement–2: $AB$ is symmetric if matrix multiplication of $A$ with $B$ is commutative

If multiplication is commutative, then $AB=BA.$

Now, $(AB)^T = B^T A^T = BA.$ Since $AB=BA$, $(AB)^T = AB.$ Hence $AB$ is symmetric.

So, Statement–2 is true.

Does Statement–2 explain Statement–1?

Statement–2 says: if $AB=BA$, then $AB$ is symmetric.

But Statement–1 concerns the matrix $ABA,$ and its symmetry follows directly from $(ABA)^T = A^T B^T A^T = ABA,$ without needing $AB=BA$.

So Statement–2 is not the correct explanation of Statement–1.

Conclusion

• Statement–1: True
• Statement–2: True
• Statement–2 is not the correct explanation for Statement–1.

Therefore, the correct option is A.