If A and B are symmetric matrices, such that AB and BA are both define

1.	If A and B are symmetric matrices, such that AB and BA are both defined, then prove that AB- BA is a skew-symmetric matrix.
Answer» Since A & B are symmetric matrices A = A’ & B = B’ To prove (AB – BA)’ = – (AB – BA) L.H.S = (AB – BA)’ (AB)’ – (BA)’ = B’A’-A’B’ = BA – AB = – (AB – BA) = R.H.S Thus proved.

If A and B are symmetric matrices, such that AB and BA are both defined, then prove that AB- BA is a skew-symmetric matrix.

Answer»

Since A & B are symmetric matrices A = A’ & B = B’

To prove (AB – BA)’ = – (AB – BA)

L.H.S = (AB – BA)’

(AB)’ – (BA)’

= B’A’-A’B’

= BA – AB

= – (AB – BA)

= R.H.S

Thus proved.

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If A and B are symmetric matrices, such that AB and BA are both defined, then prove that AB- BA is a skew-symmetric matrix.

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