Question

Assertion (A): For any symmetric matrix \( A \), \( B'A B \) is a skew-symmetric matrix.
Reason (R): A square matrix \( P \) is skew-symmetric if \( P' = -P \).

• A. Both (A) and (R) are true, and (R) is the correct explanation of (A).
• B. Both (A) and (R) are true, but (R) is not the correct explanation of (A).
• C. (A) is true, but (R) is false.
• D. (A) is false, but (R) is true.
  %Correct answer \textbf{Correct answer:}\textbf{{(A) is false, but (R) is true.}}

Hint:

Check the transpose property for the given matrices to determine symmetry or skew-symmetry.

Answer 1

The Correct Option is D

Step 1: Understanding skew-symmetric matrices.
A square matrix \( P \) is skew-symmetric if its transpose is equal to its negative, i.e., \( P' = -P \). This is stated correctly in the Reason (R). Therefore, (R) is true. Step 2: Analyze \( B'A B \) for symmetry.
Given that \( A \) is a symmetric matrix, \( A' = A \). We examine whether \( P = B'A B \) is skew-symmetric: \[ P' = (B'A B)' = B' (A') B = B' A B = P. \] Since \( P' = P \), \( B'A B \) is symmetric, not skew-symmetric. Step 3: Conclusion.
The Assertion (A) is false because \( B'A B \) is symmetric, not skew-symmetric. The Reason (R) is true as it correctly defines skew-symmetric matrices. Thus: {(A) is false, but (R) is true.}