# If A and B are symmetric matrices,such that AB and BA are both defined,then prove that $AB-BA$ is skew symmetric matrix.

we have $\left ( AB-BA \right )^{T}= \left ( AB \right )^{T}-\left ( BA \right )^{T}= B^{T}A^{T}-A^{T}B^{T}$
$= BA-AB=-\left ( AB-BA \right )$ As A & B are symmetric matrices so$A^T= A \ \ \& \ B^{T}= B$
Hence  $AB-BA$ is a skew symmetric matrix.

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