We are now tasked with solving \( A \cdot B^T + B \), where \( B^T \) is the transpose of matrix \( B \). Let’s proceed step by step.

Given matrices:

\[
A = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}
\]

### Step 1: Calculate the Transpose of Matrix \( B \)

The transpose of a matrix is obtained by flipping its rows and columns. For matrix \( B \), since it's a diagonal matrix, its transpose is the same as the original matrix:

\[
B^T = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} = B
\]

### Step 2: Perform Matrix Multiplication \( A \cdot B^T \)

Since \( B^T = B \), the matrix multiplication \( A \cdot B^T \) is the same as \( A \cdot B \), which we already computed earlier:

\[
A \cdot B^T = \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix}
\]

### Step 3: Add Matrix \( B \) to \( A \cdot B^T \)

Now, we add matrix \( B \) to the result of \( A \cdot B^T \):

\[
A \cdot B^T + B = \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix} + \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}
\]

Performing the matrix addition:

\[
A \cdot B^T + B = \begin{pmatrix} 9 & 0 \\ 0 & 9 \end{pmatrix}
\]

### Final Answer:

\[
A \cdot B^T + B = \begin{pmatrix} 9 & 0 \\ 0 & 9 \end{pmatrix}
\]

This result is the same as before since the transpose of a diagonal matrix is the matrix itself.

Would you like more clarification on matrix transposition or any other concepts?

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Here are five related questions for further exploration:
1. How would this process change if \( B \) were not a diagonal matrix?
2. What are the properties of matrix transposition in general?
3. Can you solve for \( A \cdot (B^T + B) \) instead of \( A \cdot B^T + B \)?
4. What would happen if you applied scalar multiplication to the matrices before multiplying them?
5. How does matrix transposition affect the determinant of a matrix?

**Tip:** For diagonal matrices, many operations, such as transposition and inversion, are greatly simplified. Understanding these properties can save time in larger calculations.

Learn how to solve the matrix problem A * B^T + B step-by-step. Understand matrix algebra concepts including transposition, multiplication, and addition. Suitable for advanced high school and college students.

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