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If $\mathrm{A}$ and $\mathrm{B}$ are symmetric matrices, prove that $\mathrm{AB}-\mathrm{BA}$ is a skew symmetric matrix.
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$\mathrm{A}$ and $\mathrm{B}$ are symmetric matrix is $\mathrm{A}^{\prime}=\mathrm{A}$ and $\mathrm{B}^{\prime}=\mathrm{B}$ Also $(\mathrm{AB})^{\prime}=\mathrm{B}^{\prime} \mathrm{A}^{\prime}$. Now, $(\mathrm{AB}-\mathrm{BA})^{\prime}$ $=(\mathrm{AB})^{\prime}-(\mathrm{BA})^{\prime}=\mathrm{B}^{\prime} \mathrm{A}^{\prime}-\mathrm{A}^{\prime} \mathrm{B}^{\prime}=\mathrm{BA}-\mathrm{AB}$ $=-(\mathrm{AB}-\mathrm{BA})$
Hence $\mathrm{AB}$ - BA is skew symmetric matrix.
Hence $\mathrm{AB}$ - BA is skew symmetric matrix.
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