Question

Let A and B be two symmetric matrices of same order, then which of the following statement are correct:
A. AB is symmetric
B. A+B is symmetric
C. \( A^T B = AB^T \)
D. \( BA = (AB)^T \)

• A. A, B and D only
• B. A, B and C only
• C. A, B, C and D
• D. B, C and D only

Hint:

For matrix properties, always go back to the definitions and basic transpose rules. Remember that the product of two symmetric matrices is symmetric if and only if the matrices commute. The sum of symmetric matrices is always symmetric.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A matrix M is symmetric if it is equal to its transpose, i.e., \( M^T = M \). We are given that A and B are symmetric, so \( A^T = A \) and \( B^T = B \). We need to check the validity of the given statements based on this property.

Step 2: Key Formula or Approach:
We will use the properties of transpose: \( (X+Y)^T = X^T + Y^T \) and \( (XY)^T = Y^T X^T \).

Step 3: Detailed Explanation:

A. AB is symmetric: For AB to be symmetric, we must have \( (AB)^T = AB \). Using the transpose property, \( (AB)^T = B^T A^T \). Since B and A are symmetric, \( B^T = B \) and \( A^T = A \). So, \( (AB)^T = BA \). Therefore, AB is symmetric if and only if \( BA = AB \), i.e., if A and B commute. This is not true for all symmetric matrices. So, statement A is incorrect.
B. A+B is symmetric: For A+B to be symmetric, we must have \( (A+B)^T = A+B \). Using the transpose property, \( (A+B)^T = A^T + B^T \). Since A and B are symmetric, \( A^T = A \) and \( B^T = B \). So, \( (A+B)^T = A+B \). This is always true. So, statement B is correct.
C. \( A^T B = AB^T \): We are given that A and B are symmetric, so \( A^T=A \) and \( B^T=B \). Substituting these into the statement gives \( AB = AB \). This is a tautology (always true). So, statement C is correct.
D. \( BA = (AB)^T \): Let's evaluate the right side. Using the transpose property, \( (AB)^T = B^T A^T \). Since A and B are symmetric, this becomes \( BA \). So the statement is \( BA = BA \), which is a tautology (always true). So, statement D is correct.
Step 4: Final Answer:
The correct statements are B, C, and D. Statement A is not always true.