if `A=[{:(3,-1,2),(0,5,-3),(1,-2,7):}]and B=[{:(1,0,0),(0,1,0),(0,0,1):}],`find whether AB=BA or Not .

Explanation

To determine whether \( AB = BA \) for the given matrices \( A \) and \( B \), we will perform matrix multiplication for both \( AB \) and \( BA \).

Given:
\[ A = \begin{pmatrix} 3 & -1 & 2 \\ 0 & 5 & -3 \\ 1 & -2 & 7 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

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

To find \( AB \), we multiply matrix \( A \) by matrix \( B \):
\[ AB = A \cdot B = \begin{pmatrix} 3 & -1 & 2 \\ 0 & 5 & -3 \\ 1 & -2 & 7 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

Calculating each element of \( AB \):
- First row:
- \( (3 \cdot 1) + (-1 \cdot 0) + (2 \cdot 0) = 3 \)
- \( (3 \cdot 0) + (-1 \cdot 1) + (2 \cdot 0) = -1 \)
- \( (3 \cdot 0) + (-1 \cdot 0) + (2 \cdot 1) = 2 \)

- Second row:
- \( (0 \cdot 1) + (5 \cdot 0) + (-3 \cdot 0) = 0 \)
- \( (0 \cdot 0) + (5 \cdot 1) + (-3 \cdot 0) = 5 \)
- \( (0 \cdot 0) + (5 \cdot 0) + (-3 \cdot 1) = -3 \)

- Third row:
- \( (1 \cdot 1) + (-2 \cdot 0) + (7 \cdot 0) = 1 \)
- \( (1 \cdot 0) + (-2 \cdot 1) + (7 \cdot 0) = -2 \)
- \( (1 \cdot 0) + (-2 \cdot 0) + (7 \cdot 1) = 7 \)

Thus, we have:
\[ AB = \begin{pmatrix} 3 & -1 & 2 \\ 0 & 5 & -3 \\ 1 & -2 & 7 \end{pmatrix} \]

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

Now, we calculate \( BA \):
\[ BA = B \cdot A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 3 & -1 & 2 \\ 0 & 5 & -3 \\ 1 & -2 & 7 \end{pmatrix} \]

Calculating each element of \( BA \):
- First row:
- \( (1 \cdot 3) + (0 \cdot 0) + (0 \cdot 1) = 3 \)
- \( (1 \cdot -1) + (0 \cdot 5) + (0 \cdot -2) = -1 \)
- \( (1 \cdot 2) + (0 \cdot -3) + (0 \cdot 7) = 2 \)

- Second row:
- \( (0 \cdot 3) + (1 \cdot 0) + (0 \cdot 1) = 0 \)
- \( (0 \cdot -1) + (1 \cdot 5) + (0 \cdot -2) = 5 \)
- \( (0 \cdot 2) + (1 \cdot -3) + (0 \cdot 7) = -3 \)

- Third row:
- \( (0 \cdot 3) + (0 \cdot 0) + (1 \cdot 1) = 1 \)
- \( (0 \cdot -1) + (0 \cdot 5) + (1 \cdot -2) = -2 \)
- \( (0 \cdot 2) + (0 \cdot -3) + (1 \cdot 7) = 7 \)

Thus, we have:
\[ BA = \begin{pmatrix} 3 & -1 & 2 \\ 0 & 5 & -3 \\ 1 & -2 & 7 \end{pmatrix} \]

### Conclusion

Since \( AB = BA \), we conclude that:
\[ AB = BA \]