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To solve this problem, we need to perform matrix multiplication for the given matrices \( A \) and \( B \). The matrices are defined as:

\[
A = \begin{bmatrix} 3 & 3 & -6 \\ -2 & -1 & -3 \end{bmatrix}, \quad B = \begin{bmatrix} -1 & 3 \\ -4 & -4 \\ -6 & 6 \end{bmatrix}
\]

### Step 1: Calculating \( AB \)

The resulting matrix \( AB \) will have dimensions \( 2 \times 2 \) because matrix \( A \) is \( 2 \times 3 \) and matrix \( B \) is \( 3 \times 2 \).

To find the elements of \( AB \), we perform the dot product of rows of \( A \) with columns of \( B \):

1. **Element (1,1):**  
   \[
   (3)(-1) + (3)(-4) + (-6)(-6) = -3 - 12 + 36 = 21
   \]

2. **Element (1,2):**  
   \[
   (3)(3) + (3)(-4) + (-6)(6) = 9 - 12 - 36 = -39
   \]

3. **Element (2,1):**  
   \[
   (-2)(-1) + (-1)(-4) + (-3)(-6) = 2 + 4 + 18 = 24
   \]

4. **Element (2,2):**  
   \[
   (-2)(3) + (-1)(-4) + (-3)(6) = -6 + 4 - 18 = -20
   \]

Thus, the resulting matrix \( AB \) is:
\[
AB = \begin{bmatrix} 21 & -39 \\ 24 & -20 \end{bmatrix}
\]

### Step 2: Calculating \( BA \)

The resulting matrix \( BA \) will have dimensions \( 3 \times 3 \) because matrix \( B \) is \( 3 \times 2 \) and matrix \( A \) is \( 2 \times 3 \).

To find the elements of \( BA \), we perform the dot product of rows of \( B \) with columns of \( A \):

1. **Element (1,1):**  
   \[
   (-1)(3) + (3)(-2) = -3 - 6 = -9
   \]

2. **Element (1,2):**  
   \[
   (-1)(3) + (3)(-1) = -3 - 3 = -6
   \]

3. **Element (1,3):**  
   \[
   (-1)(-6) + (3)(-3) = 6 - 9 = -3
   \]

4. **Element (2,1):**  
   \[
   (-4)(3) + (-4)(-2) = -12 + 8 = -4
   \]

5. **Element (2,2):**  
   \[
   (-4)(3) + (-4)(-1) = -12 + 4 = -8
   \]

6. **Element (2,3):**  
   \[
   (-4)(-6) + (-4)(-3) = 24 + 12 = 36
   \]

7. **Element (3,1):**  
   \[
   (-6)(3) + (6)(-2) = -18 - 12 = -30
   \]

8. **Element (3,2):**  
   \[
   (-6)(3) + (6)(-1) = -18 - 6 = -24
   \]

9. **Element (3,3):**  
   \[
   (-6)(-6) + (6)(-3) = 36 - 18 = 18
   \]

Thus, the resulting matrix \( BA \) is:
\[
BA = \begin{bmatrix} -9 & -6 & -3 \\ -4 & -8 & 36 \\ -30 & -24 & 18 \end{bmatrix}
\]

### Final Results:
\[
AB = \begin{bmatrix} 21 & -39 \\ 24 & -20 \end{bmatrix}
\]
\[
BA = \begin{bmatrix} -9 & -6 & -3 \\ -4 & -8 & 36 \\ -30 & -24 & 18 \end{bmatrix}
\]

Would you like further details or have any questions?

### Related Questions:
1. What are the properties of matrix multiplication?
2. Can matrix multiplication be commutative?
3. How do you determine if two matrices can be multiplied?
4. What is the identity matrix in the context of matrix multiplication?
5. How does matrix multiplication apply to linear transformations?

### Tip:
Always check the dimensions of matrices before performing multiplication to ensure the operation is defined.

If A = [[3, 3, -6], [-2, -1, -3]] and B = [[-1, 3], [-4, -4], [-6, 6]], find the matrices AB and BA.

This matrix multiplication problem involves multiplying two matrices A (2x3) and B (3x2) to find the resulting matrices AB (2x2) and BA (3x3). It covers key linear algebra concepts suitable for students in Grades 10-12.

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