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

(A) $AB-BA$ is a skew-symmetric matrix.
(B) $AB+BA$ is a skew-symmetric matrix.
(C) $AB^T-BA^T$ is a skew-symmetric matrix.
(D) $AB+BA$ is a symmetric matrix.

Choose the correct answer from the options given below:

(A), (C) and (D) only

The correct answer is Option (1) → (A), (C) and (D) only **

Given: $A$ and $B$ are symmetric matrices.

So, $A^{T} = A$ and $B^{T} = B$.

(A) $AB - BA$ is skew-symmetric

Check transpose:

$(AB - BA)^{T} = B^{T}A^{T} - A^{T}B^{T} = BA - AB = -(AB - BA)$

Yes, skew-symmetric.

(B) $AB + BA$ is skew-symmetric

$(AB + BA)^{T} = BA + AB = AB + BA$

It equals itself → symmetric, not skew.

(C) $AB^{T} - BA^{T}$ is skew-symmetric

Since $A$ and $B$ are symmetric: $A^{T}=A$, $B^{T}=B$.

Expression becomes: $AB - BA$ (same as part A)

Skew-symmetric → TRUE.

(D) $AB + BA$ is symmetric

From part (B), it is symmetric → TRUE.

Correct statements: (A), (C), (D).