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

 

 

 

 
 
 
 
 

Answers (1)

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 soA^T= A \ \ \& \ B^{T}= B
Hence  AB-BA is a skew symmetric matrix.

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