if `A=[{:(1,2,3),(4,5,6):}]and B=[{:(-3,-2),(0,1),(-4,-5):}],` then find AB and BA ,

Explanation

To find the products \( AB \) and \( BA \) for the given matrices \( A \) and \( B \), we will follow the steps of matrix multiplication.

### Given Matrices:
\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}, \quad B = \begin{pmatrix} -3 & -2 \\ 0 & 1 \\ -4 & -5 \end{pmatrix} \]

### Step 1: Verify Dimensions for Multiplication
- Matrix \( A \) is of dimension \( 2 \times 3 \) (2 rows and 3 columns).
- Matrix \( B \) is of dimension \( 3 \times 2 \) (3 rows and 2 columns).
- Since the number of columns in \( A \) (3) is equal to the number of rows in \( B \) (3), the product \( AB \) exists.

### Step 2: Calculate \( AB \)
To find \( AB \), we will use the formula for matrix multiplication:
\[ (AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj} \]

Calculating each element of \( AB \):

1. **Element (1,1)**:
\[ (AB)_{11} = 1 \cdot (-3) + 2 \cdot 0 + 3 \cdot (-4) = -3 + 0 - 12 = -15 \]

2. **Element (1,2)**:
\[ (AB)_{12} = 1 \cdot (-2) + 2 \cdot 1 + 3 \cdot (-5) = -2 + 2 - 15 = -15 \]

3. **Element (2,1)**:
\[ (AB)_{21} = 4 \cdot (-3) + 5 \cdot 0 + 6 \cdot (-4) = -12 + 0 - 24 = -36 \]

4. **Element (2,2)**:
\[ (AB)_{22} = 4 \cdot (-2) + 5 \cdot 1 + 6 \cdot (-5) = -8 + 5 - 30 = -33 \]

Thus, the product \( AB \) is:
\[ AB = \begin{pmatrix} -15 & -15 \\ -36 & -33 \end{pmatrix} \]

### Step 3: Verify Dimensions for \( BA \)
- Matrix \( B \) is of dimension \( 3 \times 2 \).
- Matrix \( A \) is of dimension \( 2 \times 3 \).
- Since the number of columns in \( B \) (2) is equal to the number of rows in \( A \) (2), the product \( BA \) exists.

### Step 4: Calculate \( BA \)
Calculating each element of \( BA \):

1. **Element (1,1)**:
\[ (BA)_{11} = (-3) \cdot 1 + (-2) \cdot 4 = -3 - 8 = -11 \]

2. **Element (1,2)**:
\[ (BA)_{12} = (-3) \cdot 2 + (-2) \cdot 5 = -6 - 10 = -16 \]

3. **Element (1,3)**:
\[ (BA)_{13} = (-3) \cdot 3 + (-2) \cdot 6 = -9 - 12 = -21 \]

4. **Element (2,1)**:
\[ (BA)_{21} = 0 \cdot 1 + 1 \cdot 4 = 0 + 4 = 4 \]

5. **Element (2,2)**:
\[ (BA)_{22} = 0 \cdot 2 + 1 \cdot 5 = 0 + 5 = 5 \]

6. **Element (2,3)**:
\[ (BA)_{23} = 0 \cdot 3 + 1 \cdot 6 = 0 + 6 = 6 \]

7. **Element (3,1)**:
\[ (BA)_{31} = (-4) \cdot 1 + (-5) \cdot 4 = -4 - 20 = -24 \]

8. **Element (3,2)**:
\[ (BA)_{32} = (-4) \cdot 2 + (-5) \cdot 5 = -8 - 25 = -33 \]

9. **Element (3,3)**:
\[ (BA)_{33} = (-4) \cdot 3 + (-5) \cdot 6 = -12 - 30 = -42 \]

Thus, the product \( BA \) is:
\[ BA = \begin{pmatrix} -11 & -16 & -21 \\ 4 & 5 & 6 \\ -24 & -33 & -42 \end{pmatrix} \]

### Final Results:
\[ AB = \begin{pmatrix} -15 & -15 \\ -36 & -33 \end{pmatrix}, \quad BA = \begin{pmatrix} -11 & -16 & -21 \\ 4 & 5 & 6 \\ -24 & -33 & -42 \end{pmatrix} \]