Question

If A and B are symmetric matrices, then which statements are correct?
(A) \((A-B)' = B' - A'\)
(B) \((AB+BA)\) is symmetric matrix
(C) \((AB)'= B'A'\)
(D)\( A'B' = B'A'\)
(E)\( (AB-BA) \)is skew symmetric matrix
Choose the correct answer from the options given below:

• A. (A), (C) and (E) Only
• B. (B), (D) and (E) Only
• C. (B), (C) and (E) Only
• D. (A), (B) and (E) Only

Answer 1

The Correct Option is C

To ascertain which statements are correct regarding symmetric matrices \(A\) and \(B\), analyze each given statement:

1. (A) \( (A-B)' = B' - A' \)
   Applying transpose properties: \[ (A-B)' = A' - B' \]
   For symmetric matrices, \( A' = A \) and \( B' = B \).
   Thus, \( A - B \neq B - A \). The statement is false.
2. (B) \( (AB+BA) \) is symmetric matrix
   Calculate transpose: \[ (AB+BA)' = (AB)' + (BA)' = B'A' + A'B' \]
   Since \(A\) and \(B\) are symmetric: \( A' = A \), \( B' = B \)
   \( B'A' = BA \) \text{ and } \( A'B' = AB \).
   Thus, \( (AB+BA)' = AB+BA \), so it is symmetric.
3. (C) \( (AB)'= B'A' \)
   Transpose of a product: \[ (AB)' = B'A' \] holds true for any matrices. Therefore, this is true.
4. (D) \( A'B' = B'A' \)
   For symmetric matrices: \( A' = A \) and \( B' = B \).
   But, in general, \( AB \neq BA \), thus the statement is false.
5. (E) \( (AB-BA) \) is skew symmetric matrix
   Compute the transpose: \[ (AB-BA)' = (AB)' - (BA)' = B'A' - A'B' \]
   Since \(A\) and \(B\) are symmetric: \( A' = A \) and \( B' = B \)
   \( B'A' = BA \) \text{ and } \( A'B' = AB \).
   \( (AB-BA)' = - (AB-BA) \), indicating it is skew symmetric.

Therefore, the correct statements are (B), (C), and (E). This matches the correct answer: (B), (C) and (E) Only.