Question

Which of the following is correct?

• A. \( B'AB \) is symmetric if \( A \) is symmetric.
• B. \( B'AB \) is skew-symmetric if \( A \) is symmetric.
• C. \( B'AB \) is symmetric if \( A \) is skew-symmetric.
• D. \( B'AB \) is skew-symmetric if \( A \) is skew-symmetric.

Hint:

The symmetry of the matrix transformation \( B'AB \) is determined by the symmetry properties of the matrix \( A \).

Answer 1

The Correct Option is A

Step 1: Analyze symmetry. If matrix \( A \) satisfies \( A = A' \) (is symmetric) and \( B \) is any matrix, then the expression \( (B'AB)' \) simplifies as follows: \( (B'AB)' = B'A'B' = B'AB \). This demonstrates that if \( A \) is symmetric, then \( B'AB \) is also symmetric.Step 2: Analyze skew-symmetry. If matrix \( A \) satisfies \( A = -A' \) (is skew-symmetric), then the expression \( (B'AB)' \) simplifies as follows: \( (B'AB)' = B'A'B = B'(-A)B = -B'AB \). This result shows that \( B'AB \) cannot be skew-symmetric. Conclusion:\[\boxed{\text{\( B'AB \) is symmetric if \( A \) is symmetric.}}\]