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If S is a real skew-symmetric matrix and...

If S is a real skew-symmetric matrix and det. `(I-S) ne 0`, then prove that matrix `A=(I+S) (I-S)^(-1)` is orthogonal.

A

idempotent matrix

B

symmetric matrix

C

orthogonal matrix

D

none of these

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The correct Answer is:
C
`(c )` `B=(I-A)(I+A)^(-1)`
`impliesB^(T)=(I+A^(T))^(-1)(I-A^(T))`
`=(I-A)^(-1)(I+A)`
`impliesB B^(T)=(I-A)(I+A)^(-1)(I-A)^(-1)(I+A)`
`=(I-A)(I-A)^(-1)(I+A)^(-1)(I+A)`
`=I`
(As `(I-A).(I+A)=(I+A)(I-A)`)
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