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Question
If A and B are symmetric matrices of the same order, write whether AB − BA is symmetric or skew-symmetric or neither of the two.
Solution
\[\left( AB - BA \right)^T = \left( AB \right)^T - \left( BA \right)^T \]
\[ \Rightarrow \left( AB - BA \right)^T = B^T A^T - A^T B^T \left[ \because \left( AB \right)^T = B^T A^T \right]\]
\[ \Rightarrow \left( AB - BA \right)^T = BA - AB \left[ \because B^T = \text{B and} A^T = A \right]\]
\[ \Rightarrow \left( AB - BA \right)^T = - \left( AB - BA \right)\]
Therefore, AB - BA is skew - symmetric .
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