If `A=[{:(,5,4),(,3,-1):}], B=[{:(,2,1),(,0,4):}] and C=[{:(,-3,2),(,1,0):}]`, find :
(i) A+C (ii) B-A (iii) A+B-C

Answer

To solve the given problem, we will perform matrix operations step by step.

Given matrices:
- \( A = \begin{pmatrix} 5 & 4 \\ 3 & -1 \end{pmatrix} \)
- \( B = \begin{pmatrix} 2 & 1 \\ 0 & 4 \end{pmatrix} \)
- \( C = \begin{pmatrix} -3 & 2 \\ 1 & 0 \end{pmatrix} \)

We need to find:
(i) \( A + C \)
(ii) \( B - A \)
(iii) \( A + B - C \)

### Step-by-step Solutions:

**(i) Finding \( A + C \)**

To add matrices \( A \) and \( C \), we add their corresponding elements:

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

Calculating each element:
- First element: \( 5 + (-3) = 2 \)
- Second element: \( 4 + 2 = 6 \)
- Third element: \( 3 + 1 = 4 \)
- Fourth element: \( -1 + 0 = -1 \)

Thus,

\[ A + C = \begin{pmatrix} 2 & 6 \\ 4 & -1 \end{pmatrix} \]

**(ii) Finding \( B - A \)**

To subtract matrix \( A \) from matrix \( B \), we subtract their corresponding elements:

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

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

Thus,

\[ B - A = \begin{pmatrix} -3 & -3 \\ -3 & 5 \end{pmatrix} \]

**(iii) Finding \( A + B - C \)**

First, we will find \( A + B \) and then subtract \( C \):

1. **Finding \( A + B \)**:

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

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

Thus,

\[ A + B = \begin{pmatrix} 7 & 5 \\ 3 & 3 \end{pmatrix} \]

2. **Now, subtract \( C \)**:

\[ A + B - C = \begin{pmatrix} 7 & 5 \\ 3 & 3 \end{pmatrix} - \begin{pmatrix} -3 & 2 \\ 1 & 0 \end{pmatrix} \]

Calculating each element:
- First element: \( 7 - (-3) = 7 + 3 = 10 \)
- Second element: \( 5 - 2 = 3 \)
- Third element: \( 3 - 1 = 2 \)
- Fourth element: \( 3 - 0 = 3 \)

Thus,

\[ A + B - C = \begin{pmatrix} 10 & 3 \\ 2 & 3 \end{pmatrix} \]

### Final Answers:
(i) \( A + C = \begin{pmatrix} 2 & 6 \\ 4 & -1 \end{pmatrix} \)
(ii) \( B - A = \begin{pmatrix} -3 & -3 \\ -3 & 5 \end{pmatrix} \)
(iii) \( A + B - C = \begin{pmatrix} 10 & 3 \\ 2 & 3 \end{pmatrix} \)