If A and B are symmetric matrices of the same order, show that (AB –

1.	If A and B are symmetric matrices of the same order, show that (AB – BA) is a skew symmetric matrix.
Answer» We are given that A and B are symmetric matrices of the same order then, we need to show that (AB – BA) is a skew symmetric matrix. Let us consider P is a matrix of the same order as A and B And let P = (AB – BA), we have A = A’ and B = B’ then, P’ = (AB – BA)’ P’ = ((AB)’ – (BA)’) …….using reversal law we have (CD)’=D’C’ P’ = (B’A’ – A’B’) P’ = (BA – AB) P’ = -P Hence, P is a skew symmetric matrix.

If A and B are symmetric matrices of the same order, show that (AB – BA) is a skew symmetric matrix.

Answer»

We are given that A and B are symmetric matrices of the same order then,

we need to show that (AB – BA) is a skew symmetric matrix.

Let us consider P is a matrix of the same order as A and B

And let P = (AB – BA),

we have A = A’ and B = B’

then, P’ = (AB – BA)’

P’ = ((AB)’ – (BA)’) …….using reversal law we have (CD)’=D’C’

P’ = (B’A’ – A’B’)

P’ = (BA – AB)

P’ = -P

Hence, P is a skew symmetric matrix.

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If A and B are symmetric matrices of the same order, show that (AB – BA) is a skew symmetric matrix.

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