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Let A be a skew-symmetric matrix, B= (I-...

Let A be a skew-symmetric matrix, `B= (I-A) (I+A)^(-1)`
and X and Y be column vectors conformable for
multiplication with B.
Statement-1 `(BX)^(T) (BY) = X^(T) Y`
Statement- 2 If A is skew-symmetric, then (I+A) is
non-singular.

A

Statement-1 is true, Statement -2 is true, Statement-2
is a correct explanation for Statement-1

B

Statement-1 is true, Statement-2 is true, Statement - 2
is not a correct explanation for Stamtement-1

C

Statement 1 is true, Statement - 2 is false

D

Statement-1 is false, Statement-2 is true

Text Solution

Verified by Experts

The correct Answer is:
A
`because (BX)^(T)(BY) = {(I-A)(+A)^(-1) X}^(T) (I-A) (I+A)^(-1) Y`
`=X^(T){(I+A)^(-1) }^(T) (I-A)^(T)(I-A) (I+A)^(-1) Y`
`= X^(T)(I+A^(T))^(-1) (I-A^(T))(I-A) (I+A)^(-1) Y`
`= X^(T)(I+A)^(-1) (I+A)(I-A) (I+A)^(-1) Y`
`= X^(T)(I+A)^(-1) (I-A)(I+A) (I+A)^(-1) Y`
`[because A^(T) = - A and (I-A) (I+A) = (I+A) (I-A)]`
`=X^(T) cdot Icdot IcdotIY=X^TY`
Both Statements are true, Statement-2 is correct explanation
for Statement-1.
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