Question:
If A, B are symmetric matrices of same order, then AB − BA is a
A. Skew symmetric matrix B. Symmetric matrix
C. Zero matrix D. Identity matrix
Solution:
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.