If A and B are symmetric matrices of the same order, then AB −BA is:

The correct option is Skew-symmetric matrix.

Here is the step-by-step mathematical proof explaining why this is the correct answer:

Step 1: Understand the given definitions
A square matrix $X$ is called a symmetric matrix if its transpose is equal to itself:
$X^T=X$
A square matrix $X$ is called a skew-symmetric matrix if its transpose is equal to its negative:
$X^T=-X$

Step 2: Identify the given information
We are given that $A$ and $B$ are symmetric matrices of the same order. Therefore, by definition, we have:

$$A^T=A$$

and

$$B^T=B$$

Step 3: Analyze the matrix $\mathrm{AB}-\mathrm{BA}$
To determine the type of the matrix $\mathrm{AB}-\mathrm{BA}$, let us find its transpose. Let:

$$C=\mathrm{AB}-\mathrm{BA}$$

Taking the transpose on both sides:

$$C^T={\left(\mathrm{AB}-\mathrm{BA}\right)}^T$$

Using the property of matrix transposes, ${\left(X-Y\right)}^T=X^T-Y^T$, we can write:

$$C^T={\left(\mathrm{AB}\right)}^T-{\left(\mathrm{BA}\right)}^T$$

Using the reversal law of transpose, ${\left(\mathrm{XY}\right)}^T=Y^T{X}^T$, the equation becomes:

$$C^T=B^T{A}^T-A^T{B}^T$$

Substitute the given conditions $A^T=A$ and $B^T=B$ into the expression:

$$C^T=\mathrm{BA}-\mathrm{AB}$$

Factor out a negative sign ($-$):

$$C^T=-\left(\mathrm{AB}-\mathrm{BA}\right)$$

Substituting back the original definition of $C$:

$$C^T=-C$$

Since the transpose of $\mathrm{AB}-\mathrm{BA}$ is equal to its negative, the matrix $\mathrm{AB}-\mathrm{BA}$ is a skew-symmetric matrix.