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Mathematics Question on Matrices

If AA and BB are symmetric matrices, prove that AB−BAAB−BA is a skew symmetric matrix.

Answer

It is given that AA and BB are symmetric matrices. Therefore, we have:
A′=A and B′=B....(1)A'=A\, and\, B'=B ....(1)
Now (AB−BA)′=(AB)′−(BA)′  [(A−B)′=A′−B′](AB-BA)'=(AB)'-(BA)'\,\, [(A-B)'=A'-B']
=B′A′−A′B′=B'A'-A'B'
=BA−AB[using(1)]=BA-AB [using(1)]
=−(AB−BA)=-(AB-BA)
therefore (AB−BA)′=−(AB−BA)(AB-BA)'=-(AB-BA)
Thus, (AB−BA)(AB − BA) is a skew-symmetric matrix.