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If A is a matrix of order n such that A^...

If A is a matrix of order n such that `A^(T)A=I` and X is any matrix such that `X=(A+I)^(-1) (A-I)`, then show that X is skew symmetric matrix.

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We have `X=(A+I)^(-1) (A-I)`
`:. X^(T)=((A+I)^(-1) (A-I))^(T)`
`=(A^(T)-I)(A^(T)+I)^(-1)` (1)
Now, `(A^(T)+I)(A^(T)-I)=(A^(T)-I) (A^(T)+I)`
`:. (A^(T)+I)^(-1) (A^(T)+I) (A^(T)-I) (A^(T)+I)^(-1)`
`=(A^(T)+I)^(-1) (A^(T)-I) (A^(T)+I) (A^(T)+I)^(-1)`
`implies (A^(T)-I) (A^(T)+I)^(-1)=(A^(T)+I)^(-1) (A^(T)-I)` (2)
From (1) and (2), we have
`X^(T)=(A^(T)+I)^(-1) (A^(T)-I)`
`=(A^(T)+A A^(T))^(-1) (A^(T)-A A^(T))`
`=(A^(T) (I+A))^(-1) (A^(T) (I-A))`
`=(I+A)^(-1) (A^(T))^(-1) (A^(T) (I-A))`
`=(I+A)^(-1) ((A^(-1))^(T)A^(T)) (I-A)`
`=(I+A)^(-1) (I-A)`
`=-(A+I)^(-1) (A-I)`
`=-X`
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