Given `A=[{:(,1,4),(,2,3):}] and B=[{:(,-4,-1),(,-3,-2):}]`
(i) find the matrix 2A+B.
(ii) find a matrix C such that : `C+B=[{:(,0,0),(,0,0):}]`

Answer

Let's solve the problem step by step.

### Given Matrices:
- Matrix A:
\[ A = \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} \]

- Matrix B:
\[ B = \begin{pmatrix} -4 & -1 \\ -3 & -2 \end{pmatrix} \]

### (i) Find the matrix \(2A + B\)

**Step 1: Multiply matrix A by 2.**
\[ 2A = 2 \times \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 2 \times 1 & 2 \times 4 \\ 2 \times 2 & 2 \times 3 \end{pmatrix} = \begin{pmatrix} 2 & 8 \\ 4 & 6 \end{pmatrix} \]

**Step 2: Add matrix \(2A\) to matrix \(B\).**
\[ 2A + B = \begin{pmatrix} 2 & 8 \\ 4 & 6 \end{pmatrix} + \begin{pmatrix} -4 & -1 \\ -3 & -2 \end{pmatrix} = \begin{pmatrix} 2 + (-4) & 8 + (-1) \\ 4 + (-3) & 6 + (-2) \end{pmatrix} = \begin{pmatrix} -2 & 7 \\ 1 & 4 \end{pmatrix} \]

### Answer for (i):
\[ 2A + B = \begin{pmatrix} -2 & 7 \\ 1 & 4 \end{pmatrix} \]

### (ii) Find a matrix \(C\) such that \(C + B = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\)

**Step 1: Rearrange the equation to find \(C\).**
\[ C + B = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \implies C = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} - B \]

**Step 2: Substitute the values of matrix \(B\).**
\[ C = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} - \begin{pmatrix} -4 & -1 \\ -3 & -2 \end{pmatrix} = \begin{pmatrix} 0 - (-4) & 0 - (-1) \\ 0 - (-3) & 0 - (-2) \end{pmatrix} = \begin{pmatrix} 4 & 1 \\ 3 & 2 \end{pmatrix} \]

### Answer for (ii):
\[ C = \begin{pmatrix} 4 & 1 \\ 3 & 2 \end{pmatrix} \]

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