Question

If $A$ and $B$ are two symmetric matrices of order $3$.

Statement I $A\left(BA\right)$ and $\left(AB\right)A$ are symmetric matrices.

Statement II $AB$ is a symmetric matrix If matrix multiplication of $A$ and $B$ is commutative.

• A. Statement I is correct, Statement II is correct; Statement II is not a correct explanation for the statement I
• B. Statement I is correct, Statement II is incorrect
• C. Statement I is incorrect, Statement II is correct
• D. Statement I is correct, Statement II is correct, Statement II is the correct explanation for Statement I

Answer 1

The correct option is D

Statement I is correct, Statement II is correct, Statement II is the correct explanation for Statement I

Explanation for the correct option:

$\because A$ and $B$ are symmetric matrices

$\Rightarrow A^T=A$ and $B^T=B$

$\begin{array}{rcl}{[A(BA\left)\right]}^T& =& {\left(BA\right)}^T{A}^T\\ & =& \left(A^T{B}^T\right)A^T\\ & =& \left(AB\right)A\\ & =& A\left(BA\right)\end{array}$

$\therefore$$A\left(BA\right)$ is symmetric.

Similarly, $\left(AB\right)A$ is also symmetric.

$\therefore$Both statement I and statement II are correct.

Hence, Option ‘D’ is Correct.