If `A=[(2,-1),(4,2)],B=[(4,3),(-2,1)]` and `C=[(-2,-3),(-1,2)]`, find each of the following :
(i) `2B+3C`
(ii) `-2A+(B+C)`
(iii) `(2A-3B)-C`
(iv) `A+(2B-C)`
(v) `A+(B+C)`
(vi) `(A+B)+C`.

Answer

To solve the given problem, we will perform matrix operations step by step for each part of the question.

Given matrices:
- \( A = \begin{pmatrix} 2 & -1 \\ 4 & 2 \end{pmatrix} \)
- \( B = \begin{pmatrix} 4 & 3 \\ -2 & 1 \end{pmatrix} \)
- \( C = \begin{pmatrix} -2 & -3 \\ -1 & 2 \end{pmatrix} \)

### (i) Calculate \( 2B + 3C \)

1. **Calculate \( 2B \)**:
\[ 2B = 2 \times \begin{pmatrix} 4 & 3 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} 8 & 6 \\ -4 & 2 \end{pmatrix} \]

2. **Calculate \( 3C \)**:
\[ 3C = 3 \times \begin{pmatrix} -2 & -3 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} -6 & -9 \\ -3 & 6 \end{pmatrix} \]

3. **Add \( 2B \) and \( 3C \)**:
\[ 2B + 3C = \begin{pmatrix} 8 & 6 \\ -4 & 2 \end{pmatrix} + \begin{pmatrix} -6 & -9 \\ -3 & 6 \end{pmatrix} = \begin{pmatrix} 8 - 6 & 6 - 9 \\ -4 - 3 & 2 + 6 \end{pmatrix} = \begin{pmatrix} 2 & -3 \\ -7 & 8 \end{pmatrix} \]

### (ii) Calculate \( -2A + (B + C) \)

1. **Calculate \( B + C \)**:
\[ B + C = \begin{pmatrix} 4 & 3 \\ -2 & 1 \end{pmatrix} + \begin{pmatrix} -2 & -3 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} 4 - 2 & 3 - 3 \\ -2 - 1 & 1 + 2 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ -3 & 3 \end{pmatrix} \]

2. **Calculate \( -2A \)**:
\[ -2A = -2 \times \begin{pmatrix} 2 & -1 \\ 4 & 2 \end{pmatrix} = \begin{pmatrix} -4 & 2 \\ -8 & -4 \end{pmatrix} \]

3. **Add \( -2A \) and \( (B + C) \)**:
\[ -2A + (B + C) = \begin{pmatrix} -4 & 2 \\ -8 & -4 \end{pmatrix} + \begin{pmatrix} 2 & 0 \\ -3 & 3 \end{pmatrix} = \begin{pmatrix} -4 + 2 & 2 + 0 \\ -8 - 3 & -4 + 3 \end{pmatrix} = \begin{pmatrix} -2 & 2 \\ -11 & -1 \end{pmatrix} \]

### (iii) Calculate \( (2A - 3B) - C \)

1. **Calculate \( 2A \)**:
\[ 2A = 2 \times \begin{pmatrix} 2 & -1 \\ 4 & 2 \end{pmatrix} = \begin{pmatrix} 4 & -2 \\ 8 & 4 \end{pmatrix} \]

2. **Calculate \( 3B \)**:
\[ 3B = 3 \times \begin{pmatrix} 4 & 3 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} 12 & 9 \\ -6 & 3 \end{pmatrix} \]

3. **Calculate \( 2A - 3B \)**:
\[ 2A - 3B = \begin{pmatrix} 4 & -2 \\ 8 & 4 \end{pmatrix} - \begin{pmatrix} 12 & 9 \\ -6 & 3 \end{pmatrix} = \begin{pmatrix} 4 - 12 & -2 - 9 \\ 8 + 6 & 4 - 3 \end{pmatrix} = \begin{pmatrix} -8 & -11 \\ 14 & 1 \end{pmatrix} \]

4. **Subtract \( C \)**:
\[ (2A - 3B) - C = \begin{pmatrix} -8 & -11 \\ 14 & 1 \end{pmatrix} - \begin{pmatrix} -2 & -3 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} -8 + 2 & -11 + 3 \\ 14 + 1 & 1 - 2 \end{pmatrix} = \begin{pmatrix} -6 & -8 \\ 15 & -1 \end{pmatrix} \]

### (iv) Calculate \( A + (2B - C) \)

1. **Calculate \( 2B \)** (already calculated):
\[ 2B = \begin{pmatrix} 8 & 6 \\ -4 & 2 \end{pmatrix} \]

2. **Subtract \( C \)**:
\[ 2B - C = \begin{pmatrix} 8 & 6 \\ -4 & 2 \end{pmatrix} - \begin{pmatrix} -2 & -3 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} 8 + 2 & 6 + 3 \\ -4 + 1 & 2 - 2 \end{pmatrix} = \begin{pmatrix} 10 & 9 \\ -3 & 0 \end{pmatrix} \]

3. **Add \( A \)**:
\[ A + (2B - C) = \begin{pmatrix} 2 & -1 \\ 4 & 2 \end{pmatrix} + \begin{pmatrix} 10 & 9 \\ -3 & 0 \end{pmatrix} = \begin{pmatrix} 2 + 10 & -1 + 9 \\ 4 - 3 & 2 + 0 \end{pmatrix} = \begin{pmatrix} 12 & 8 \\ 1 & 2 \end{pmatrix} \]

### (v) Calculate \( A + (B + C) \)

1. **Use \( B + C \)** (already calculated):
\[ B + C = \begin{pmatrix} 2 & 0 \\ -3 & 3 \end{pmatrix} \]

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

### (vi) Calculate \( (A + B) + C \)

1. **Calculate \( A + B \)**:
\[ A + B = \begin{pmatrix} 2 & -1 \\ 4 & 2 \end{pmatrix} + \begin{pmatrix} 4 & 3 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} 2 + 4 & -1 + 3 \\ 4 - 2 & 2 + 1 \end{pmatrix} = \begin{pmatrix} 6 & 2 \\ 2 & 3 \end{pmatrix} \]

2. **Add \( C \)**:
\[ (A + B) + C = \begin{pmatrix} 6 & 2 \\ 2 & 3 \end{pmatrix} + \begin{pmatrix} -2 & -3 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} 6 - 2 & 2 - 3 \\ 2 - 1 & 3 + 2 \end{pmatrix} = \begin{pmatrix} 4 & -1 \\ 1 & 5 \end{pmatrix} \]

### Summary of Results:
1. \( 2B + 3C = \begin{pmatrix} 2 & -3 \\ -7 & 8 \end{pmatrix} \)
2. \( -2A + (B + C) = \begin{pmatrix} -2 & 2 \\ -11 & -1 \end{pmatrix} \)
3. \( (2A - 3B) - C = \begin{pmatrix} -6 & -8 \\ 15 & -1 \end{pmatrix} \)
4. \( A + (2B - C) = \begin{pmatrix} 12 & 8 \\ 1 & 2 \end{pmatrix} \)
5. \( A + (B + C) = \begin{pmatrix} 4 & -1 \\ 1 & 5 \end{pmatrix} \)
6. \( (A + B) + C = \begin{pmatrix} 4 & -1 \\ 1 & 5 \end{pmatrix} \)