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Question
If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix.
Solution
If A and B are symmetric matrices.
∴ A’ = A and B’ = B
(AB - BA) = (AB)’ - (BA)’ [∵ (X - Y) = X’ - Y’]
= B’A’ - A’B’ [∵ (XY) =Y’X’]
= BA - AB [∵ B’ = B, A’ = A]
= -(AB - BA)
∴ AB - BA is a skew symmetric matrix.
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