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

Answer

A

Explanation

Since, A and B are symmetric matrices
`therefore A' = A and B' = B`
Statement - 1 Let `P = A (BA)`
`therefore P' = (A(BA))' = (BA)' A'`
`=(A'B')A'`
`=(AB) A [because A' = A, B' = B]`
`= A (BA)`[By associative law]
`rArr A (BA)` is symmetric
Now, let `Q= (AB)A`
`Q' = ((AB)A)'`
`= A' (AB)' = A' (B'A')`
` = A(BA) [because A' = A, B' = B]`
`=(AB)A` [By associative law]
`= Q`
`rArr (AB)A ` is symmetric
` therefore` Statment- 1 is true.
Statement - 2 `(AB)'=B'A'=BA " "[becauseA'=A,B'=B]`
`=AB [because AB= BA]`
`rArr AB` is symmetric matrix
`therefore ` Staement -2 is true.
Hence, both Statement are true, Statement - 2 is not a correct
explanation for Statement-1