Question

Let A and B be two symmetic matrices of order 3.
Statement - 1 : $A\left(BA\right)$ and $\left(AB\right)A$ 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
• 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 1

Answer: (B)

Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1

Consider, $(A\left(BA\right){)}^T$
$=(BA{)}^T{A}^T$
$=\left(A^T{B}^T\right)A^T$
$=\left(AB\right)A$ ($\because$ A and B are symmetric)
$=A\left(BA\right)$
$\Rightarrow \left(A\right(BA{)}^T)=A(BA)$
Hence, A(BA) is symmetric.
Now, $\left(\right(AB)A{)}^T=A^T(AB{)}^T$
$=A^T\left(B^T{A}^T\right)$
$=A\left(BA\right)$
$=\left(AB\right)A$
$\Rightarrow \left(\right(AB)A{)}^T=(AB)A$
Hence, $\left(AB\right)A$ is symmetric
Statement 1 is true.
Now, $(AB{)}^T=B^T{A}^T$
$=BA$
$=AB$ (given matrix A and B satisfies commutative property)
$\Rightarrow (AB{)}^T=AB$
Hence, statement 2 is true.
But it is not the correct explanantion of statement 1