If `A=[{:(,2),(,5):}], B=[{:(,1),(,4):}] and C=[{:(,6),(,-2):}]`, find
(i) B+C (ii) A-C
(iii) A+B-C (iv) A-B+C

Answer

To solve the problem step by step, we will perform the matrix operations as specified in the question.

Given matrices:
- \( A = \begin{pmatrix} 2 & 5 \end{pmatrix} \)
- \( B = \begin{pmatrix} 1 & 4 \end{pmatrix} \)
- \( C = \begin{pmatrix} 6 & -2 \end{pmatrix} \)

### (i) Find \( B + C \)

To add matrices \( B \) and \( C \):
\[ B + C = \begin{pmatrix} 1 & 4 \end{pmatrix} + \begin{pmatrix} 6 & -2 \end{pmatrix} \]
We add corresponding elements:
\[ = \begin{pmatrix} 1 + 6 & 4 + (-2) \end{pmatrix} = \begin{pmatrix} 7 & 2 \end{pmatrix} \]

### (ii) Find \( A - C \)

To subtract matrix \( C \) from \( A \):
\[ A - C = \begin{pmatrix} 2 & 5 \end{pmatrix} - \begin{pmatrix} 6 & -2 \end{pmatrix} \]
We subtract corresponding elements:
\[ = \begin{pmatrix} 2 - 6 & 5 - (-2) \end{pmatrix} = \begin{pmatrix} -4 & 7 \end{pmatrix} \]

### (iii) Find \( A + B - C \)

First, we add \( A \) and \( B \):
\[ A + B = \begin{pmatrix} 2 & 5 \end{pmatrix} + \begin{pmatrix} 1 & 4 \end{pmatrix} \]
Adding corresponding elements:
\[ = \begin{pmatrix} 2 + 1 & 5 + 4 \end{pmatrix} = \begin{pmatrix} 3 & 9 \end{pmatrix} \]
Now, we subtract \( C \):
\[ A + B - C = \begin{pmatrix} 3 & 9 \end{pmatrix} - \begin{pmatrix} 6 & -2 \end{pmatrix} \]
Subtracting corresponding elements:
\[ = \begin{pmatrix} 3 - 6 & 9 - (-2) \end{pmatrix} = \begin{pmatrix} -3 & 11 \end{pmatrix} \]

### (iv) Find \( A - B + C \)

First, we subtract \( B \) from \( A \):
\[ A - B = \begin{pmatrix} 2 & 5 \end{pmatrix} - \begin{pmatrix} 1 & 4 \end{pmatrix} \]
Subtracting corresponding elements:
\[ = \begin{pmatrix} 2 - 1 & 5 - 4 \end{pmatrix} = \begin{pmatrix} 1 & 1 \end{pmatrix} \]
Now, we add \( C \):
\[ A - B + C = \begin{pmatrix} 1 & 1 \end{pmatrix} + \begin{pmatrix} 6 & -2 \end{pmatrix} \]
Adding corresponding elements:
\[ = \begin{pmatrix} 1 + 6 & 1 + (-2) \end{pmatrix} = \begin{pmatrix} 7 & -1 \end{pmatrix} \]

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