Let $A$ and $B$ be two symmetric matrices of same order. Then, the matrix $A B-B A$ is

Let $A$ and $B$ be two symmetric matrices of same order. Then, the matrix $A B-B A$ is
  1. a symmetric matrix
  2. a skew-symmetric matrix
  3. a null matrix
  4. the identity matrix

Solution

Given, $A=A^{\prime}, B=B^{\prime}$ Now, $(A B-B A)^{\prime}=(A B)^{\prime}-(B A)^{\prime}$ $\begin{aligned} & =B^{\prime} A^{\prime}-A^{\prime} B^{\prime} \\ & =B A-A B \\ & =-(A B-B A) \end{aligned}$ $\therefore A B-B A$ is a skew-symmetric matrix.

Asked in: AP EAMCET 2009

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