Question:
If \(A\) and \(B\) are symmetric matrices, prove that \(AB−BA\) is a skew symmetric matrix.
If \(A\) and \(B\) are symmetric matrices, prove that \(AB−BA\) is a skew symmetric matrix.
Updated On: Sep 21, 2023
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Solution and Explanation
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.
\(A'=A\, and\, B'=B ....(1)\)
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.
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