If A and B are symmetric matrices of the same order, write whether AB

1.	If A and B are symmetric matrices of the same order, write whether AB – BA is symmetric or skew-symmetric or neither of the two.
Answer» A and B are symmetric matrices, ∴ A’ = A and B’ = B …(i) Consider, (AB – BA)’ = (AB)’ – (BA)’ [(a – b)’ = a’ – b’] = B’A’ – A’B’ [(AB)’ = B’A’] = BA – AB [from (i)] = - (AB – BA) ∴ (AB – BA)’ = - (AB – BA) Hence, (AB – BA) is a skew symmetric matrix.

If A and B are symmetric matrices of the same order, write whether AB – BA is symmetric or skew-symmetric or neither of the two.

Answer»

A and B are symmetric matrices,

∴ A’ = A and B’ = B …(i)

Consider,

(AB – BA)’ = (AB)’ – (BA)’ [(a – b)’ = a’ – b’]

= B’A’ – A’B’ [(AB)’ = B’A’]

= BA – AB [from (i)]

= - (AB – BA)

∴ (AB – BA)’ = - (AB – BA)

Hence,

(AB – BA) is a skew symmetric matrix.

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If A and B are symmetric matrices of the same order, write whether AB – BA is symmetric or skew-symmetric or neither of the two.

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