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To solve the problem \( A + B^T \), where matrices \( A \) and \( B \) are defined as:

\[
A = \begin{pmatrix} 7 & -1 & 0 \\ -1 & 7 & -2 \\ 0 & -2 & 7 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 1 & 2 \\ 1 & -5 & 1 \\ 2 & 1 & -4 \end{pmatrix}
\]

we first need to compute the transpose of matrix \( B \) (denoted \( B^T \)) and then add it to matrix \( A \).

### Step 1: Calculate \( B^T \)
The transpose of \( B \) is obtained by swapping the rows and columns of \( B \):

\[
B^T = \begin{pmatrix} 3 & 1 & 2 \\ 1 & -5 & 1 \\ 2 & 1 & -4 \end{pmatrix}^T = \begin{pmatrix} 3 & 1 & 2 \\ 1 & -5 & 1 \\ 2 & 1 & -4 \end{pmatrix}
\]

### Step 2: Add \( A \) and \( B^T \)

Now, we add the matrices \( A \) and \( B^T \) element-wise:

\[
A + B^T = \begin{pmatrix} 7 & -1 & 0 \\ -1 & 7 & -2 \\ 0 & -2 & 7 \end{pmatrix} + \begin{pmatrix} 3 & 1 & 2 \\ 1 & -5 & 1 \\ 2 & 1 & -4 \end{pmatrix}
\]

Calculating each element:

- Row 1: \( 7 + 3 = 10 \), \( -1 + 1 = 0 \), \( 0 + 2 = 2 \)
- Row 2: \( -1 + 1 = 0 \), \( 7 + (-5) = 2 \), \( -2 + 1 = -1 \)
- Row 3: \( 0 + 2 = 2 \), \( -2 + 1 = -1 \), \( 7 + (-4) = 3 \)

Thus:

\[
A + B^T = \begin{pmatrix} 10 & 0 & 2 \\ 0 & 2 & -1 \\ 2 & -1 & 3 \end{pmatrix}
\]

### Final Answer
\[
A + B^T = \begin{pmatrix} 10 & 0 & 2 \\ 0 & 2 & -1 \\ 2 & -1 & 3 \end{pmatrix}
\]

Would you like a detailed explanation of any of these steps, or do you have any further questions?

Here are some related questions to deepen your understanding:
1. What is the significance of matrix transposition in linear algebra?
2. How would you calculate the product \( AB \) of the matrices \( A \) and \( B \)?
3. Can you determine if matrix \( A \) is symmetric? How?
4. How does matrix addition affect properties like symmetry or diagonal dominance?
5. What are some real-world applications of matrix addition and transposition?

**Tip:** In matrix operations, be cautious with the order of operations, especially when dealing with transpositions, as they can significantly alter the matrix's structure.

Calculate \( A + B^T \), where \( A \) and \( B \) are defined as:

\[
A = \begin{pmatrix} 7 & -1 & 0 \\ -1 & 7 & -2 \\ 0 & -2 & 7 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 1 & 2 \\ 1 & -5 & 1 \\ 2 & 1 & -4 \end{pmatrix}
\]

Learn how to calculate \( A + B^T \) with matrices \( A \) and \( B \). This problem covers matrix addition, transposition, and matrix properties. Ideal for students in Grades 11-12.

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