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?

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.