If A and B are symmetric matrices of the same order, then which of the following are true?

(A) AB - BA is a skew symmetric matrix
(B) AB is a symmetric matrix
(C) AB is a scalar matrix
(D) AB + BA is a symmetric matrix

Choose the correct answer from the options given below:

(A) and (D) only

The correct answer is Option (2) → (A) and (D) only

Given: $A$ and $B$ are symmetric matrices of the same order, i.e., $A^T = A$ and $B^T = B$

Check (A): $(AB - BA)^T = B^T A^T - A^T B^T = BA - AB = - (AB - BA)$

⟹ $AB - BA$ is skew-symmetric

⟹ (A) is true

Check (B): In general, $AB$ is not symmetric unless $A$ and $B$ commute, i.e., $AB = BA$

⟹ (B) is false

Check (C): No condition suggests $AB$ is a scalar matrix

⟹ (C) is false

Check (D): $(AB + BA)^T = B^T A^T + A^T B^T = BA + AB = AB + BA$

⟹ $AB + BA$ is symmetric

⟹ (D) is true