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Question
If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
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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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