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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 false
Statement $-1$ is false, Statement $-2$ is true
Statement $-1$ is true, Statement $-2$ is true; Statement $-2$ is not acorrect explanation for Statement $-1$
Statement $-1$ is true, Statement $-2$ is true; Statement $-2$ is a correct explanation for Statement $-1$
Solution
$\therefore A^{\prime}=A$
$B^{\prime}=B$
Now $(\mathrm{A}(\mathrm{BA}))^{\prime}=(\mathrm{BA})^{\prime} \mathrm{A}^{\prime}$
$=\left(A^{\prime} B^{\prime}\right) A^{\prime}=(A B) A=A(B A)$
Similarly $((\mathrm{AB}) \mathrm{A})^{\prime}=(\mathrm{AB}) \mathrm{A}$
So, $A(B A)$ and $(A B) A$ are symmetric matrices.
Again $(\mathrm{AB})^{\prime}=\mathrm{B}^{\prime} \mathrm{A}^{\prime}=\mathrm{BA}$
Now if $\mathrm{BA}=\mathrm{AB},$ then $\mathrm{AB}$ is symmetric matrix.
