If `A=[{:(1,1,-1),(2,-3,4),(3,-2,3):}]and B=[{:(-1,-2,-1),(6,12,6),(5,10,5):}]` then which of the following is/are correct ?
1. A and B commute.
2. AB is a null matrix.
Select the correct answer using the code given below :

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To solve the problem, we need to analyze the matrices \( A \) and \( B \) and check the two statements provided.

Given:
\[ A = \begin{pmatrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} -1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5 \end{pmatrix} \]

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

To find \( AB \), we will multiply matrix \( A \) by matrix \( B \).

\[ AB = \begin{pmatrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{pmatrix} \begin{pmatrix} -1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5 \end{pmatrix} \]

Calculating each element of the resulting matrix:

- **First row, first column**:
\[ 1 \cdot (-1) + 1 \cdot 6 + (-1) \cdot 5 = -1 + 6 - 5 = 0 \]

- **First row, second column**:
\[ 1 \cdot (-2) + 1 \cdot 12 + (-1) \cdot 10 = -2 + 12 - 10 = 0 \]

- **First row, third column**:
\[ 1 \cdot (-1) + 1 \cdot 6 + (-1) \cdot 5 = -1 + 6 - 5 = 0 \]

- **Second row, first column**:
\[ 2 \cdot (-1) + (-3) \cdot 6 + 4 \cdot 5 = -2 - 18 + 20 = 0 \]

- **Second row, second column**:
\[ 2 \cdot (-2) + (-3) \cdot 12 + 4 \cdot 10 = -4 - 36 + 40 = 0 \]

- **Second row, third column**:
\[ 2 \cdot (-1) + (-3) \cdot 6 + 4 \cdot 5 = -2 - 18 + 20 = 0 \]

- **Third row, first column**:
\[ 3 \cdot (-1) + (-2) \cdot 6 + 3 \cdot 5 = -3 - 12 + 15 = 0 \]

- **Third row, second column**:
\[ 3 \cdot (-2) + (-2) \cdot 12 + 3 \cdot 10 = -6 - 24 + 30 = 0 \]

- **Third row, third column**:
\[ 3 \cdot (-1) + (-2) \cdot 6 + 3 \cdot 5 = -3 - 12 + 15 = 0 \]

Thus, we find:
\[ AB = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]

### Step 2: Check if \( A \) and \( B \) commute

Next, we need to calculate \( BA \).

\[ BA = \begin{pmatrix} -1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5 \end{pmatrix} \begin{pmatrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{pmatrix} \]

Calculating each element of the resulting matrix:

- **First row, first column**:
\[ -1 \cdot 1 + (-2) \cdot 2 + (-1) \cdot 3 = -1 - 4 - 3 = -8 \]

- **First row, second column**:
\[ -1 \cdot 1 + (-2) \cdot (-3) + (-1) \cdot (-2) = -1 + 6 + 2 = 7 \]

- **First row, third column**:
\[ -1 \cdot (-1) + (-2) \cdot 4 + (-1) \cdot 3 = 1 - 8 - 3 = -10 \]

- **Second row, first column**:
\[ 6 \cdot 1 + 12 \cdot 2 + 6 \cdot 3 = 6 + 24 + 18 = 48 \]

- **Second row, second column**:
\[ 6 \cdot 1 + 12 \cdot (-3) + 6 \cdot (-2) = 6 - 36 - 12 = -42 \]

- **Second row, third column**:
\[ 6 \cdot (-1) + 12 \cdot 4 + 6 \cdot 3 = -6 + 48 + 18 = 60 \]

- **Third row, first column**:
\[ 5 \cdot 1 + 10 \cdot 2 + 5 \cdot 3 = 5 + 20 + 15 = 40 \]

- **Third row, second column**:
\[ 5 \cdot 1 + 10 \cdot (-3) + 5 \cdot (-2) = 5 - 30 - 10 = -35 \]

- **Third row, third column**:
\[ 5 \cdot (-1) + 10 \cdot 4 + 5 \cdot 3 = -5 + 40 + 15 = 50 \]

Thus, we find:
\[ BA = \begin{pmatrix} -8 & 7 & -10 \\ 48 & -42 & 60 \\ 40 & -35 & 50 \end{pmatrix} \]

### Conclusion

1. Since \( AB \) is a null matrix, statement 2 is correct.
2. Since \( AB \neq BA \), statement 1 is incorrect.

**Final Answer**: The correct option is **2 only**.