`if A'=[{:(3,4),(-1,2),(0,1):}]and B=[{:(-1,2,1),(1,2,3):}],`then verify that `(i) (A+B)'=A'+B'(ii) (A-B)'=A'-B'`

Explanation

To verify the given statements, we will follow these steps:

### Given:
- \( A' = \begin{pmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{pmatrix} \)
- \( B = \begin{pmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{pmatrix} \)

### Step 1: Find \( A \)
Since \( A' \) is the transpose of \( A \), we can find \( A \) by transposing \( A' \):
\[ A = (A')' = \begin{pmatrix} 3 & -1 & 0 \\ 4 & 2 & 1 \end{pmatrix} \]

### Step 2: Find \( B' \)
Now, we find the transpose of \( B \):
\[ B' = \begin{pmatrix} -1 & 1 \\ 2 & 2 \\ 1 & 3 \end{pmatrix} \]

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

#### Step 3.1: Calculate \( A + B \)
To add matrices \( A \) and \( B \), we need to ensure they have the same dimensions. Here, \( A \) is \( 2 \times 3 \) and \( B \) is \( 2 \times 3 \):
\[ A + B = \begin{pmatrix} 3 & -1 & 0 \\ 4 & 2 & 1 \end{pmatrix} + \begin{pmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{pmatrix} = \begin{pmatrix} 3 - 1 & -1 + 2 & 0 + 1 \\ 4 + 1 & 2 + 2 & 1 + 3 \end{pmatrix} = \begin{pmatrix} 2 & 1 & 1 \\ 5 & 4 & 4 \end{pmatrix} \]

#### Step 3.2: Find \( (A + B)' \)
Now, we take the transpose of \( A + B \):
\[ (A + B)' = \begin{pmatrix} 2 & 1 & 1 \\ 5 & 4 & 4 \end{pmatrix}' = \begin{pmatrix} 2 & 5 \\ 1 & 4 \\ 1 & 4 \end{pmatrix} \]

#### Step 3.3: Calculate \( A' + B' \)
Now we add \( A' \) and \( B' \):
\[ A' + B' = \begin{pmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} -1 & 1 \\ 2 & 2 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} 3 - 1 & 4 + 1 \\ -1 + 2 & 2 + 2 \\ 0 + 1 & 1 + 3 \end{pmatrix} = \begin{pmatrix} 2 & 5 \\ 1 & 4 \\ 1 & 4 \end{pmatrix} \]

### Conclusion for Part (i)
Since \( (A + B)' = A' + B' \), we have verified the first statement.

### Step 4: Verify \( (A - B)' = A' - B' \)

#### Step 4.1: Calculate \( A - B \)
\[ A - B = \begin{pmatrix} 3 & -1 & 0 \\ 4 & 2 & 1 \end{pmatrix} - \begin{pmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{pmatrix} = \begin{pmatrix} 3 + 1 & -1 - 2 & 0 - 1 \\ 4 - 1 & 2 - 2 & 1 - 3 \end{pmatrix} = \begin{pmatrix} 4 & -3 & -1 \\ 3 & 0 & -2 \end{pmatrix} \]

#### Step 4.2: Find \( (A - B)' \)
Now, we take the transpose of \( A - B \):
\[ (A - B)' = \begin{pmatrix} 4 & -3 & -1 \\ 3 & 0 & -2 \end{pmatrix}' = \begin{pmatrix} 4 & 3 \\ -3 & 0 \\ -1 & -2 \end{pmatrix} \]

#### Step 4.3: Calculate \( A' - B' \)
Now we subtract \( B' \) from \( A' \):
\[ A' - B' = \begin{pmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{pmatrix} - \begin{pmatrix} -1 & 1 \\ 2 & 2 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} 3 + 1 & 4 - 1 \\ -1 - 2 & 2 - 2 \\ 0 - 1 & 1 - 3 \end{pmatrix} = \begin{pmatrix} 4 & 3 \\ -3 & 0 \\ -1 & -2 \end{pmatrix} \]

### Conclusion for Part (ii)
Since \( (A - B)' = A' - B' \), we have verified the second statement.

### Final Result
Both statements are verified:
1. \( (A + B)' = A' + B' \)
2. \( (A - B)' = A' - B' \)