Let A and B two symmetric 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 with B is commutative.

Options

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

Answer

B

Explanation

We have,
`(A(BA))^T=(BA)^TA^T=(A^TB^T)A^T=(AB)A=A(BA)`
and,` ((AB)A)^T=A^T(AB)^T=A^T(B^TA^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^TA^T=BA=AB`
`rArr`AB is symmetric matrix.
So, statement -2 is true. But, it is not a correct explanation for statement -1.