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

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Question

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

Theorem
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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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Chapter 3: Matrices - Miscellaneous Exercise on Chapter 3 [Page 72]

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NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 3 Matrices
Miscellaneous Exercise on Chapter 3 | Q 1. | Page 72

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