Question

If \(A\) and \(B\) are symmetric matrices, prove that \(AB−BA\) is a skew symmetric matrix.

Answer 1

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