A =[2 0 0 2]  e B = [3 0 0 3]. SOlve A *B + B.

Given the matrices \( A \) and \( B \):

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

We need to solve \( A \cdot B + B \).

### Step 1: Matrix Multiplication \( A \cdot B \)

To calculate the product of two matrices, we multiply each element of the rows of matrix \( A \) by the corresponding elements of the columns of matrix \( B \).

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

Performing the matrix multiplication:

\[
A \cdot B = \begin{pmatrix} (2 \cdot 3) + (0 \cdot 0) & (2 \cdot 0) + (0 \cdot 3) \\ (0 \cdot 3) + (2 \cdot 0) & (0 \cdot 0) + (2 \cdot 3) \end{pmatrix} = \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix}
\]

### Step 2: Matrix Addition \( (A \cdot B) + B \)

Now, add matrix \( B \) to the result of \( A \cdot B \).

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

Performing the addition:

\[
(A \cdot B) + B = \begin{pmatrix} 6 + 3 & 0 + 0 \\ 0 + 0 & 6 + 3 \end{pmatrix} = \begin{pmatrix} 9 & 0 \\ 0 & 9 \end{pmatrix}
\]

### Final Answer:

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

Would you like to explore any details further or ask related questions?

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Here are five related questions for further exploration:
1. What happens if we change the values of \( A \) or \( B \)?
2. How would the result change if the matrices were not diagonal?
3. Can you describe the properties of diagonal matrices like \( A \) and \( B \)?
4. What happens when we multiply three matrices instead of two?
5. How does matrix multiplication differ from element-wise multiplication?

**Tip:** Diagonal matrices simplify many matrix operations, as only the diagonal elements are involved in the calculations.

Learn how to solve the matrix operation A * B + B using diagonal matrices A and B. This problem covers matrix multiplication, matrix addition, and the properties of diagonal matrices. Suitable for advanced high school students.

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