Which of the following are correct?

(A) If A and B are symmetric matrices such that AB = BA, then AB is symmetric.
(B) If A and B are symmetric matrices of the same order, then (A + B) is a symmetric matrix.
(C) If A and B are symmetric matrices of the same order, then (AB - BA) is a symmetric matrix.
(D) If A and B are symmetric matrices of the same order, then (AB + BA) is a skew symmetric matrix.

Choose the correct answer from the options given below:

(A) and (B) only

The correct answer is Option (1) → (A) and (B) only

$\text{(A) }A,B\ \text{symmetric and }AB=BA$

$(AB)^{T}=B^{T}A^{T}=BA=AB\ \Rightarrow\ AB\ \text{symmetric. True.}$

$\text{(B) }A,B\ \text{symmetric of same order}$

$(A+B)^{T}=A^{T}+B^{T}=A+B\ \Rightarrow\ \text{symmetric. True.}$

$\text{(C) }A,B\ \text{symmetric}$

$(AB-BA)^{T}=B^{T}A^{T}-A^{T}B^{T}=BA-AB=-(AB-BA)\ \Rightarrow\ \text{skew-symmetric, not symmetric. False.}$

$\text{(D) }A,B\ \text{symmetric}$

$(AB+BA)^{T}=B^{T}A^{T}+A^{T}B^{T}=BA+AB=AB+BA\ \Rightarrow\ \text{symmetric, not skew-symmetric. False.}$