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To solve the given problem, we need to perform the transpose operation on each matrix and then add the resulting matrices.

The two matrices given in the question are:

\[
\text{Matrix A} = \begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix}
\quad \text{and} \quad
\text{Matrix B} = \begin{bmatrix} -1 & 3 \\ 2 & -1 \end{bmatrix}.
\]

### Step 1: Transpose Both Matrices

The transpose of a matrix is obtained by swapping its rows with its columns.

1. Transposing Matrix A:

\[
\text{Transpose of A} = \begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix}^\top = \begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix}.
\]

(Note: Transposing Matrix A does not change it since it's already symmetric.)

2. Transposing Matrix B:

\[
\text{Transpose of B} = \begin{bmatrix} -1 & 3 \\ 2 & -1 \end{bmatrix}^\top = \begin{bmatrix} -1 & 2 \\ 3 & -1 \end{bmatrix}.
\]

### Step 2: Add the Transposed Matrices

Now, we add the transposed matrices:

\[
\begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix} + \begin{bmatrix} -1 & 2 \\ 3 & -1 \end{bmatrix} = \begin{bmatrix} 2 + (-1) & -1 + 2 \\ -1 + 3 & 3 + (-1) \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix}.
\]

### Step 3: Compare with the Options

The resulting matrix is:

\[
\begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix}.
\]

This matches option B.

### Answer:
**B. \(\begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix}\)**

Would you like more details or have any questions? Here are some related questions:

1. How do you determine the transpose of a matrix?
2. What is the effect of transposing a symmetric matrix?
3. How do you perform matrix addition?
4. What are some properties of matrix operations (like addition and transpose)?
5. How can you verify your results in matrix operations?

**Tip:** Always double-check the order of operations when dealing with matrices, especially when transposing and adding.

Transpose of matrices (2 -1, -1 3) + (-1 3, 2 -1) is?

Learn how to solve matrix operations involving transposition and addition. This problem demonstrates the steps for finding the transpose of two matrices and adding them, covering key matrix concepts for students in Grades 10-12.

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