Question

Let \(S\) and \(T\) be symmetric matrices of the same order, which of the following statements are true?
• [A.] \(S + T\) is a symmetric matrix
• [B.] \(ST\) is a symmetric matrix
• [C.] \(ST = TS\) is a symmetric matrix
• [D.] \(ST - TS\) is a symmetric matrix

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

Hint:

Matrix multiplication is not commutative; symmetry of product depends on commutativity.

Answer 1

The Correct Option is B

Concept: If \(A\) is symmetric, then \(A^T = A\).

Step 1: Check A. \[ (S+T)^T = S^T + T^T = S + T \] So symmetric ⇒ TRUE.

Step 2: Check B. \[ (ST)^T = T^T S^T = TS \] Not equal to \(ST\) generally ⇒ FALSE.

Step 3: Check C. If \(ST = TS\), then: \[ (ST)^T = TS = ST \] Thus symmetric ⇒ TRUE.

Step 4: Check D. \[ (ST - TS)^T = TS - ST = -(ST - TS) \] Not equal ⇒ not symmetric ⇒ FALSE.

Step 5: Final answer. \[ \boxed{(2)\ A,\ C\ \text{only}} \]