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.