Step 1: Definition of Symmetric Matrix:
A matrix M is symmetric if its transpose equals itself (\( M^T = M \)). Given symmetric matrices A and B, we know \( A^T = A \) and \( B^T = B \). The task is to validate the given statements based on this.
Step 2: Transpose Properties:
Key properties used: \( (X+Y)^T = X^T + Y^T \) and \( (XY)^T = Y^T X^T \).
Step 3: Statement Analysis:
A. AB is symmetric:
To check if AB is symmetric, verify \( (AB)^T = AB \). Using transpose rules, \( (AB)^T = B^T A^T \). Since A and B are symmetric, \( B^T = B \) and \( A^T = A \), thus \( (AB)^T = BA \). AB is symmetric ONLY if \( BA = AB \) (A and B commute). This isn't universally true; therefore, statement A is false.
B. A+B is symmetric:
To check if A+B is symmetric, verify \( (A+B)^T = A+B \). Using transpose rules, \( (A+B)^T = A^T + B^T \). Because A and B are symmetric, \( A^T = A \) and \( B^T = B \). Thus, \( (A+B)^T = A+B \). This is always true; statement B is correct.
C. \( A^T B = AB^T \):
Given A and B are symmetric, so \( A^T=A \) and \( B^T=B \). Substituting into the statement yields \( AB = AB \). This is always true (a tautology); therefore, statement C is correct.
D. \( BA = (AB)^T \):
Evaluate the right side: \( (AB)^T = B^T A^T \). Since A and B are symmetric, this simplifies to \( BA \). The statement becomes \( BA = BA \), which is a tautology (always true). So, statement D is correct.
Step 4: Conclusion:
Statements B, C, and D are correct. Statement A is not always true.