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To multiply the matrices \( A \) and \( B \), we need to perform matrix multiplication, which involves taking the dot product of rows from \( A \) with columns from \( B \).

### Given matrices:
Matrix \( A \):
\[
A = \begin{bmatrix} -1 & -2 & 1 & -2 \\ 3 & 2 & 3 & 0 \\ -2 & 1 & 1 & -2 \end{bmatrix}
\]

Matrix \( B \):
\[
B = \begin{bmatrix} 1 & 2 & -2 \\ 1 & -2 & 3 \\ 2 & -2 & -2 \\ 1 & 0 & 0 \end{bmatrix}
\]

### Dimensions:
- Matrix \( A \) is \( 3 \times 4 \).
- Matrix \( B \) is \( 4 \times 3 \).

The resulting matrix \( C = A \times B \) will have dimensions \( 3 \times 3 \) (since \( A \) has 3 rows and \( B \) has 3 columns).

### Step-by-Step Calculation:
Let’s calculate each element in \( C \) by taking the dot product of each row of \( A \) with each column of \( B \).

1. **Element \( C_{11} \)**:
   \[
   C_{11} = (-1 \times 1) + (-2 \times 1) + (1 \times 2) + (-2 \times 1) = -1 - 2 + 2 - 2 = -3
   \]

2. **Element \( C_{12} \)**:
   \[
   C_{12} = (-1 \times 2) + (-2 \times -2) + (1 \times -2) + (-2 \times 0) = -2 + 4 - 2 + 0 = 0
   \]

3. **Element \( C_{13} \)**:
   \[
   C_{13} = (-1 \times -2) + (-2 \times 3) + (1 \times -2) + (-2 \times 0) = 2 - 6 - 2 + 0 = -6
   \]

4. **Element \( C_{21} \)**:
   \[
   C_{21} = (3 \times 1) + (2 \times 1) + (3 \times 2) + (0 \times 1) = 3 + 2 + 6 + 0 = 11
   \]

5. **Element \( C_{22} \)**:
   \[
   C_{22} = (3 \times 2) + (2 \times -2) + (3 \times -2) + (0 \times 0) = 6 - 4 - 6 + 0 = -4
   \]

6. **Element \( C_{23} \)**:
   \[
   C_{23} = (3 \times -2) + (2 \times 3) + (3 \times -2) + (0 \times 0) = -6 + 6 - 6 + 0 = -6
   \]

7. **Element \( C_{31} \)**:
   \[
   C_{31} = (-2 \times 1) + (1 \times 1) + (1 \times 2) + (-2 \times 1) = -2 + 1 + 2 - 2 = -1
   \]

8. **Element \( C_{32} \)**:
   \[
   C_{32} = (-2 \times 2) + (1 \times -2) + (1 \times -2) + (-2 \times 0) = -4 - 2 - 2 + 0 = -8
   \]

9. **Element \( C_{33} \)**:
   \[
   C_{33} = (-2 \times -2) + (1 \times 3) + (1 \times -2) + (-2 \times 0) = 4 + 3 - 2 + 0 = 5
   \]

### Resulting Matrix \( C \):
\[
C = \begin{bmatrix} -3 & 0 & -6 \\ 11 & -4 & -6 \\ -1 & -8 & 5 \end{bmatrix}
\]

Would you like further explanation or details on matrix multiplication?

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Here are some questions for further exploration:

1. How do you perform matrix multiplication with matrices of different sizes?
2. Why is matrix multiplication not commutative?
3. How can the properties of matrices simplify complex calculations?
4. What are some practical applications of matrix multiplication in data science?
5. How do special matrices (like identity or zero matrices) interact with other matrices?

**Tip:** Always check the dimensions of matrices before multiplying to ensure they are compatible for the operation.

Multiply the matrices \( A = \begin{bmatrix} -1 & -2 & 1 & -2 \\ 3 & 2 & 3 & 0 \\ -2 & 1 & 1 & -2 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 2 & -2 \\ 1 & -2 & 3 \\ 2 & -2 & -2 \\ 1 & 0 & 0 \end{bmatrix} \).

Explore the process of multiplying two matrices, A and B, with dimensions 3x4 and 4x3, respectively. This solution includes a detailed, step-by-step calculation of each element in the resulting 3x3 matrix. Ideal for students in Grades 10-12 studying linear algebra.

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