Let $A$ and $B$ be two symmetric matrices of order 3 . This question has Statement $-1$ and Statement $-2$.…
Let $A$ and $B$ be two symmetric matrices of order 3 .
This question has Statement $-1$ and Statement $-2$. Of the four choices given after the statements, choose the one that best describes the two statements.
Statement $-1$ : $\mathrm{A}(\mathrm{BA})$ and $(\mathrm{AB}) \mathrm{A}$ are symmetric matrices.
Statement - 2 : $\quad A B$ is symmetric matrix if matrix multiplication of $A$ and $B$ 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 $-1$
Solution
$
\begin{aligned}
& A^{\top}=A, B^{\top}=B \\
& (A(B A))^{\top}=(B A)^{\top} A^{\top}=\left(A^{\top} B^{\top}\right) A=(A B) A=A(B A) \\
& ((A B) A)^{\top}=A^{\top}(A B)^{\top}=A\left(B^{\top} A^{\top}\right)=A(B A)=(A B) A
\end{aligned}
$
$\therefore$ Statement $-1$ is correct
Statement - 2
$
(A B)^{\top}=B^{\top} A^{\top}=B A=A B
$
( $\because \mathrm{AB}$ is commutative)
Statement $-2$ is also correct but it is not correct explanation of Statement $-1$