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If A is a skew symmetric matrix, then B=...

If `A` is a skew symmetric matrix, then `B=(I-A)(I+A)^(-1)` is (where `I` is an identity matrix of same order as of `A`)

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We have `A=(I+S)(I-S)^(-1)`
`:. A^(T)=[(I-S)^(-1)]^(T) (I+S)^(T)=[(I-S)^(T)]^(-1) (I+S)^(T)`
But `(I-S)^(T)=I^(T)-S^(T)=I +S" "("as "S^(T)=-S)`
and `(I+S)^(T)=I^(T)+S^(T)=I-S`
`:. A^(T)=(I+S)^(-1) (I-S)`
`:. A^(T)A=(I+S)^(-1) (I-S) (I+S) (I-S)^(-1)`
`=(I+S)^(-1) (I+S) (I-S) (I-S)^(-1)`
`=I`
Thus, a is orthogonal.
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CENGAGE-MATRICES-Exercise 13.5
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  6. If A=[[cos alpha, -sin alpha] , [sin alpha, cos alpha]], B=[[cos2beta,...

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  7. If A=[(1,2,2),(2,2,3),(1,-1,3)], C=[(2,1,1),(2,2,1),(1,1,1)], D=[(10),...

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  8. If A is a 2xx2 matrix such that A^(2)-4A+3I=O, then prove that (A+3I)^...

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  10. Prove that inverse of a skew-symmetric matrix (if it exists) is skew-s...

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  12. If A is a skew symmetric matrix, then B=(I-A)(I+A)^(-1) is (where I is...

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  13. Prove that ("adj. "A)^(-1)=("adj. "A^(-1)).

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  15. Show that the two matrices A, P^(-1) AP have the same characteristic r...

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