Let A and B be two symmetric matrices of order 3. Statement-1 : A(BA) and (AB)A are symmetric matrices.
Let A and B be 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.
(a) Statement-1 is true, Statement-2 is false.
(b) Statement-1 is false, Statement-2 is true.
(c) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(d) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
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(d) : Let A(BA) = P
Then PT = (ABA)T = ATBTAT (Transversal rule)
= ABA = P
Thus P is symmetric.
Again, A(BA) = (AB)A by associativity.
Also (AB)T = BTAT = BA = AB
(Q A and B are commutative)
⇒ AB is also symmetric.
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