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

Hint:

To determine if a matrix is symmetric or skew-symmetric, check the transpose property. A matrix is skew-symmetric if its transpose equals its negative, and symmetric if it equals its transpose.

Answer 1

The Correct Option is D

Step 1: Understanding skew-symmetric matrices.
A matrix \( P \) is called skew-symmetric if its transpose is equal to its negative, i.e., \( P' = -P \). This property is correctly stated in the Reason (R), meaning that \( P \) must satisfy the condition \( P' = -P \). Therefore, the Reason (R) is true. Step 2: Analyze \( B'A B \) for symmetry.
Let’s assume that \( A \) is a symmetric matrix, so we have \( A' = A \). Now we investigate whether \( P = B'A B \) is skew-symmetric. To do this, we calculate the transpose of \( P \): \[ P' = (B'A B)' = B' (A') B = B' A B. \] Since \( A' = A \), we get: \[ P' = B' A B = P. \] This shows that \( P' = P \), meaning that \( P \) is symmetric, not skew-symmetric. Step 3: Conclusion.
Since \( P = B'A B \) is symmetric and not skew-symmetric, the Assertion (A) is false. However, the Reason (R) correctly describes the property of skew-symmetric matrices, so it is true. Hence: \[ \boxed{\text{(A) is false, but (R) is true.}} \]