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 matix.

(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 matrix is symmetric if it equals its own transpose: $A^T = A$

For symmetric matrices A and B: $A^T = A$ and $B^T = B$

Transpose rule: $(AB)^T = B^T A^T$

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Statement (A): If AB = BA and both are symmetric, then AB is symmetric.

$(AB)^T = B^T A^T$

$= BA$

$= AB$ (given that AB = BA)

Therefore $(AB)^T = AB$, so AB is symmetric.

Statement (A) is correct.

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Statement (B): (A+B) is symmetric.

$(A+B)^T = A^T + B^T$

$= A + B$

Therefore $(A+B)^T = (A+B)$, so (A+B) is symmetric.

Statement (B) is correct.

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Statement (C): (AB-BA) is symmetric.

$(AB-BA)^T = (AB)^T - (BA)^T$

$= B^T A^T - A^T B^T$

$= BA - AB$

$= -(AB - BA)$

Therefore $(AB-BA)^T = -(AB-BA)$, so (AB-BA) is skew-symmetric, not symmetric.

Statement (C) is incorrect.

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Statement (D): (AB+BA) is skew-symmetric.

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

$= B^T A^T + A^T B^T$

$= BA + AB$

$= AB + BA$

Therefore $(AB+BA)^T = (AB+BA)$, so (AB+BA) is symmetric, not skew-symmetric.

Statement (D) is incorrect.

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Only statements (A) and (B) are correct.