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: Investigate symmetry properties.
Let \( A \) be a symmetric matrix, i.e., \( A = A' \), and let \( B \) be any matrix.
Now, consider the expression: \[ (B'AB)' = B'A'B' = B'AB. \]
This shows that \( B'AB \) is symmetric if \( A \) is symmetric. Step 2: Investigate skew-symmetry.
If \( A \) is skew-symmetric, i.e., \( A = -A' \): \[ (B'AB)' = B'A'B = B'(-A)B = -B'AB. \]
This indicates that \( B'AB \) cannot be skew-symmetric. Final Answer: \[ \boxed{\text{\( B'AB \) is symmetric if \( A \) is symmetric.}} \]