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Statement-1 (Assertion and Statement- 2 ...

Statement-1 (Assertion and Statement- 2 (Reason)
Each of these questions also has four alternative
choices, only one of which is the correct answer. You
have to select the correct choice as given below.
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- is true, Statement -2 is true, Statement-2
is a correct explanation for Statement-5

B

Statement-1 is true, Statement-2 is true, Sttatement - 2
is not a correct explanation for Stamtement-5

C

Statement 1 is true, Statement - 2 is false

D

Statement-1 is false, Statement-2 is true

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the two statements regarding the skew-symmetric matrix \( A \) and the derived matrix \( B \). ### Step 1: Understanding the Definitions A skew-symmetric matrix \( A \) satisfies the property \( A^T = -A \). The matrix \( B \) is defined as: \[ B = (I - A)(I + A)^{-1} \] where \( I \) is the identity matrix. ### Step 2: Proving Statement 1 We need to show that: \[ (BX)^T (BY) = X^T Y \] Substituting the expression for \( B \): \[ (BX)^T = (I - A)(I + A)^{-1}X^T \] \[ (BY) = (I - A)(I + A)^{-1}Y \] Now, we can write: \[ (BX)^T (BY) = ((I - A)(I + A)^{-1}X)^T (I - A)(I + A)^{-1}Y \] Using the property of transpose: \[ = X^T ((I + A)^{-1})^T (I - A)^T (I - A)(I + A)^{-1}Y \] Since \( (I + A)^{-1} \) is the inverse of \( I + A \), we have: \[ = X^T (I + A)^{-T} (I - A)^T (I - A)(I + A)^{-1}Y \] Using the property of the transpose of a product: \[ = X^T (I + A^T)^{-1} (I - A^T)(I - A)(I + A)^{-1}Y \] Since \( A^T = -A \): \[ = X^T (I - A)^{-1} (I + A)(I - A)(I + A)^{-1}Y \] This simplifies to: \[ = X^T Y \] Thus, we have shown that: \[ (BX)^T (BY) = X^T Y \] ### Step 3: Proving Statement 2 We need to show that \( (I + A) \) is non-singular if \( A \) is skew-symmetric. To show that \( (I + A) \) is non-singular, we can use the property of determinants. Since \( A \) is skew-symmetric, its eigenvalues are either zero or purely imaginary. Thus, \( 1 + \lambda_i \) (where \( \lambda_i \) are the eigenvalues of \( A \)) will not be zero for any eigenvalue \( \lambda_i \) of \( A \). Hence, \( (I + A) \) is non-singular. ### Conclusion Both statements are true: - Statement 1 is true since \( (BX)^T (BY) = X^T Y \). - Statement 2 is true since \( (I + A) \) is non-singular for a skew-symmetric matrix \( A \). Thus, the correct choice is: **A: Both statements are true, and Statement 2 is the correct reason for Statement 1.**
To solve the problem, we need to analyze the two statements regarding the skew-symmetric matrix \( A \) and the derived matrix \( B \). ### Step 1: Understanding the Definitions A skew-symmetric matrix \( A \) satisfies the property \( A^T = -A \). The matrix \( B \) is defined as: \[ B = (I - A)(I + A)^{-1} \] where \( I \) is the identity matrix. ...
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