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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-1
Consider, (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)A
Hence, (AB)A is symmetric
Statement 1 is true.
Now, (AB)T=BTAT
=BA
=AB (given matrix A and B satisfies commutative property)
(AB)T=AB
Hence, statement 2 is true.
But it is not the correct explanantion of statement 1

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