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

If A is a matrix of order n such that `A^(T)A=I` and X is any matric s

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

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