If A and B are symmetric matrices then AB

Engineering

Mathematics

Special Type of Square Matrices

Symmetric and Skew Symmetric Matrices

Question

If A and B are symmetric matrices then AB – BA is a _______.

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AB – BA is a skew symmetric matrix.
A and B are symmetric matrices.

∴ A' = A, B' = B

Now, [AB – BA]' = (AB)' – (BA)'

= B'A' – A'B' ⋯⋯⋅ [∵ (AB)' = B'A']

= BA – AB ⋯⋅ {∵ A' = A and B' = B}

= –[AB – BA]

So, [AB – BA] is a skew symmetric matrix.