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.

Question

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 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

Tardigrade Admin · Accepted Answer

∴ A' = A B' = B Now (A(BA))' = (BA)'A' = (A'B')A' = (AB)A = A(BA) Similarly ((AB)A)' = (AB)A So, A(BA) and (AB)A are symmetric matrices. Again (AB)' = B'A' = BA Now if BA = AB, then AB is symmetric matrix.

Q. 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.

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

Statement-1 is true, Statement-2 is false

Statement-1 is false, Statement-2 is true

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

$∴$ A' = A
B' = B
Now (A(BA))' = (BA)'A'
= (A'B')A' = (AB)A = A(BA)
Similarly ((AB)A)' = (AB)A
So, A(BA) and (AB)A are symmetric matrices.
Again (AB)' = B'A' = BA
Now if BA = AB, then AB is symmetric matrix.

Q. 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.