Question

Let $A$ and $B$ be two symmetric 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 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.

Given, $A=A$
$B=B$
Now $\left(A\right(BA\left)\right)=\left(BA\right)A=\left(AB\right)A=\left(AB\right)A$
$=A\left(BA\right)$
Similarly $\left(\right(AB\left)A\right)=\left(AB\right)A$
So, $A\left(BA\right)$ and $\left(AB\right)A$ are symmetric matrices.
Again $\left(AB\right)=BA$
Now if $BA=AB$, then $AB$ is symmetric matrix.
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
Hence, option 'B' is correct.