Let A and B be two symmetic matrices of order 3.

Statement-1: A (BA) and (AB) A are symmetric matrices.

Statement-2: AB is symmetric matrix if matrix multiplication of A and B is commutative.

Statement-1 is True, Statement-2 is True; Statement-2 is not a correct explanation for Statement-1.

The correct answer is Option 2: Statement-1 is True, Statement-2 is True; Statement-2 is not a correct explanation for Statement-1.

We have,

$(A(BA))^T = (BA)^T A^T =(A^T B^T) A^T =(AB) A = A (BA)$

and, $((AB) A)^T = A^T (AB)^T = A^T (B^T A^T) = A (BA) = (AB) A$

$∴A (BA)$ and $(AB) A$ are symmetric matrices.

So, statement-1 is true.

If matrix multiplication of A and B is commutative. Then,

$AB = BA$

$∴(AB)^T = B^T A^T = BA = AB$

$⇒ AB$ is symmetric matrix.

Is Statement 2 the correct explanation for Statement 1?

Statement 2 is not the correct explanation for Statement 1. While Statement 2 correctly says that the product AB will be symmetric only when AB=BA  (i.e., when A and B commute), Statement 1 does not depend on this condition at all. In Statement 1, the matrices are of the form A(BA) and (AB)A, which simplify to ABA. This expression remains symmetric because A and B themselves are symmetric, regardless of whether they commute or not. Therefore, even if AB≠BA, Statement 1 still holds true. Hence, Statement 2, although correct on its own, does not explain Statement 1