`if A=[{:(1,2,-3),(5,0,2),(1,-1,1):}],B=[{:(3,-1,2),(4,2,5),(2,0,3):}]and c=[{:(4,1,2),(0,3,2),(1,-2,3):}],` then compure (A+B) and (B-C), Also , verify that A+(B-C)=(A+B)-C.`

Explanation

To solve the given problem, we will follow these steps:

### Step 1: Compute \( A + B \)

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

To compute \( A + B \), we add the corresponding elements of matrices \( A \) and \( B \):

\[ A + B = \begin{pmatrix} 1 + 3 & 2 + (-1) & -3 + 2 \\ 5 + 4 & 0 + 2 & 2 + 5 \\ 1 + 2 & -1 + 0 & 1 + 3 \end{pmatrix} \]

Calculating each element:
- First row: \( 1 + 3 = 4 \), \( 2 - 1 = 1 \), \( -3 + 2 = -1 \)
- Second row: \( 5 + 4 = 9 \), \( 0 + 2 = 2 \), \( 2 + 5 = 7 \)
- Third row: \( 1 + 2 = 3 \), \( -1 + 0 = -1 \), \( 1 + 3 = 4 \)

Thus,
\[ A + B = \begin{pmatrix} 4 & 1 & -1 \\ 9 & 2 & 7 \\ 3 & -1 & 4 \end{pmatrix} \]

### Step 2: Compute \( B - C \)

Given matrix:
\[ C = \begin{pmatrix} 4 & 1 & 2 \\ 0 & 3 & 2 \\ 1 & -2 & 3 \end{pmatrix} \]

To compute \( B - C \), we subtract the corresponding elements of matrix \( C \) from matrix \( B \):

\[ B - C = \begin{pmatrix} 3 - 4 & -1 - 1 & 2 - 2 \\ 4 - 0 & 2 - 3 & 5 - 2 \\ 2 - 1 & 0 - (-2) & 3 - 3 \end{pmatrix} \]

Calculating each element:
- First row: \( 3 - 4 = -1 \), \( -1 - 1 = -2 \), \( 2 - 2 = 0 \)
- Second row: \( 4 - 0 = 4 \), \( 2 - 3 = -1 \), \( 5 - 2 = 3 \)
- Third row: \( 2 - 1 = 1 \), \( 0 + 2 = 2 \), \( 3 - 3 = 0 \)

Thus,
\[ B - C = \begin{pmatrix} -1 & -2 & 0 \\ 4 & -1 & 3 \\ 1 & 2 & 0 \end{pmatrix} \]

### Step 3: Verify \( A + (B - C) = (A + B) - C \)

Now we need to compute \( A + (B - C) \) and \( (A + B) - C \).

#### Compute \( A + (B - C) \)

Using \( A \) and \( B - C \):
\[ A + (B - C) = \begin{pmatrix} 1 & 2 & -3 \\ 5 & 0 & 2 \\ 1 & -1 & 1 \end{pmatrix} + \begin{pmatrix} -1 & -2 & 0 \\ 4 & -1 & 3 \\ 1 & 2 & 0 \end{pmatrix} \]

Calculating:
- First row: \( 1 - 1 = 0 \), \( 2 - 2 = 0 \), \( -3 + 0 = -3 \)
- Second row: \( 5 + 4 = 9 \), \( 0 - 1 = -1 \), \( 2 + 3 = 5 \)
- Third row: \( 1 + 1 = 2 \), \( -1 + 2 = 1 \), \( 1 + 0 = 1 \)

Thus,
\[ A + (B - C) = \begin{pmatrix} 0 & 0 & -3 \\ 9 & -1 & 5 \\ 2 & 1 & 1 \end{pmatrix} \]

#### Compute \( (A + B) - C \)

Using \( A + B \) and \( C \):
\[ (A + B) - C = \begin{pmatrix} 4 & 1 & -1 \\ 9 & 2 & 7 \\ 3 & -1 & 4 \end{pmatrix} - \begin{pmatrix} 4 & 1 & 2 \\ 0 & 3 & 2 \\ 1 & -2 & 3 \end{pmatrix} \]

Calculating:
- First row: \( 4 - 4 = 0 \), \( 1 - 1 = 0 \), \( -1 - 2 = -3 \)
- Second row: \( 9 - 0 = 9 \), \( 2 - 3 = -1 \), \( 7 - 2 = 5 \)
- Third row: \( 3 - 1 = 2 \), \( -1 - (-2) = 1 \), \( 4 - 3 = 1 \)

Thus,
\[ (A + B) - C = \begin{pmatrix} 0 & 0 & -3 \\ 9 & -1 & 5 \\ 2 & 1 & 1 \end{pmatrix} \]

### Conclusion

Since \( A + (B - C) = (A + B) - C \), we have verified the required property.