Question

# 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.

A
Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
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B
Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1
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C
Statement-1 is True, Statement-2 is False.
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D
Statement-1 is False, Statement-2 is True.
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Solution

## The correct option is D Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1Consider, (A(BA))T =(BA)TAT =(ATBT)AT =(AB)A (∵ A and B are symmetric) =A(BA)⇒(A(BA)T)=A(BA)Hence, A(BA) is symmetric.Now, ((AB)A)T=AT(AB)T =AT(BTAT) =A(BA) =(AB)A⇒((AB)A)T=(AB)AHence, (AB)A is symmetricStatement 1 is true.Now, (AB)T=BTAT =BA =AB (given matrix A and B satisfies commutative property) ⇒(AB)T=ABHence, statement 2 is true.But it is not the correct explanantion of statement 1

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