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 symmetric matrix satisfies $A^T = A$. For transpose operations, $(AB)^T = B^T A^T$ (order reverses).

Given that $A$ and $B$ are symmetric matrices of the same order, we have $A^T = A$ and $B^T = B$.

---

For option (A), checking if $AB - BA$ is skew symmetric:

A matrix $M$ is skew symmetric if $M^T = -M$.

$(AB - BA)^T$

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

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

$= BA - AB$

$= -(AB - BA)$

Therefore, $AB - BA$ is skew symmetric. Option (A) is true.

---

For option (B), checking if $AB$ is symmetric:

$(AB)^T$

$= B^T A^T$

$= BA$

For $AB$ to be symmetric, we need $AB = BA$.

However, two symmetric matrices do not always commute.

Counterexample: $A = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 1 \\ 1 & 0 \end{bmatrix}$

Both are symmetric, but $AB \neq BA$.

Option (B) is false.

---

For option (C), checking if $AB$ is a scalar matrix:

A scalar matrix has the form $kI$ where $I$ is the identity matrix.

The product of two symmetric matrices does not generally produce a scalar matrix.

Option (C) is false.

---

For option (D), checking if $AB + BA$ is symmetric:

$(AB + BA)^T$

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

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

$= BA + AB$

$= AB + BA$

Therefore, $AB + BA$ is symmetric. Option (D) is true.

---

Only statements (A) and (D) are true.