Let A and B be two symmetric matrices of order 3.<br>Statement-1 : A(BA) and (AB)A are symmetric matrices.<br>Statement-2 : AB is symmetric matrix if matrix multiplication of A with B is commutative.
(A) Statement-1 is true, Statement-2 is false.
(B) Statement-1 is true, Statement-2 is true
(C) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for statement-1.
(D) Statement-1 is false, Statement-2 is true.
Step-by-Step Solution
Key Concept: A matrix M is symmetric if M^T = M. Given A^T = A and B^T = B. For Statement-1: (ABA)^T = A^T B^T A^T = ABA, so ABA is symmetric. Similarly, (A(BA))^T = (ABA)^T = ABA, and ((AB)A)^T = A^T(AB)^T = A(B^T A^T) = ABA. For Statement-2: If AB is symmetric, then (AB)^T = AB, which means B^T A^T = AB, so BA = AB. This is the condition for commutativity.
Statement-1: Since A and B are symmetric, A^T = A and B^T = B. Consider (ABA)^T = A^T B^T A^T = ABA. Thus, ABA is symmetric. Since A(BA) = ABA and (AB)A = ABA, both are symmetric. Statement-1 is true. Statement-2: If AB is symmetric, then (AB)^T = AB, which implies B^T A^T = AB. Since A and B are symmetric, BA = AB. Thus, if AB is symmetric, then AB = BA. Statement-2 is also true. However, Statement-2 is not the definition of why Statement-1 is true, but rather a property related to the symmetry of the product.
Correct Answer: 4