If A, B are symmetric matrices of same order, then AB − BA is a


If AB are symmetric matrices of same order, then AB − BA is a

A. Skew symmetric matrix B. Symmetric matrix

C. Zero matrix D. Identity matrix


The correct answer is A.

A and B are symmetric matrices, therefore, we have:

$A^{\prime}=A$ and $B^{\prime}=B$     .........(1)

$\begin{aligned} \text { Consider }(A B-B A)^{\prime} &=(A B)^{\prime}-(B A)^{\prime} & &\left[(A-B)^{\prime}=A^{\prime}-B^{\prime}\right] \\ &=B^{\prime} A^{\prime}-A^{\prime} B^{\prime} & &\left[(A B)^{\prime}=B^{\prime} A^{\prime}\right] \\ &=B A-A B & &[\text { by }(1)] \\ &=-(A B-B A) & & \end{aligned}$

$\therefore(A B-B A)^{\prime}=-(A B-B A)$

Thus, $(A B-B A)$ is a skew-symmetric matrix.


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