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.