Given `A=[{:(,2,-6),(,2,0):}], B=[{:(,-3,2),(,4,0):}] and C=[{:(,4,0),(,0,2):}]`. Find the matrix X such that A+2X=2B+C.

Answer

To solve the equation \( A + 2X = 2B + C \), we will follow these steps:

### Step 1: Write down the matrices
Given:
- \( A = \begin{pmatrix} 2 & -6 \\ 2 & 0 \end{pmatrix} \)
- \( B = \begin{pmatrix} -3 & 2 \\ 4 & 0 \end{pmatrix} \)
- \( C = \begin{pmatrix} 4 & 0 \\ 0 & 2 \end{pmatrix} \)

### Step 2: Calculate \( 2B \)
To find \( 2B \):
\[ 2B = 2 \times \begin{pmatrix} -3 & 2 \\ 4 & 0 \end{pmatrix} = \begin{pmatrix} 2 \times -3 & 2 \times 2 \\ 2 \times 4 & 2 \times 0 \end{pmatrix} = \begin{pmatrix} -6 & 4 \\ 8 & 0 \end{pmatrix} \]

### Step 3: Calculate \( 2B + C \)
Now, we add \( 2B \) and \( C \):
\[ 2B + C = \begin{pmatrix} -6 & 4 \\ 8 & 0 \end{pmatrix} + \begin{pmatrix} 4 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} -6 + 4 & 4 + 0 \\ 8 + 0 & 0 + 2 \end{pmatrix} = \begin{pmatrix} -2 & 4 \\ 8 & 2 \end{pmatrix} \]

### Step 4: Rearranging the equation
From the equation \( A + 2X = 2B + C \), we can rearrange it to find \( 2X \):
\[ 2X = (2B + C) - A \]

### Step 5: Calculate \( 2X \)
Now we subtract \( A \) from \( 2B + C \):
\[ 2X = \begin{pmatrix} -2 & 4 \\ 8 & 2 \end{pmatrix} - \begin{pmatrix} 2 & -6 \\ 2 & 0 \end{pmatrix} = \begin{pmatrix} -2 - 2 & 4 - (-6) \\ 8 - 2 & 2 - 0 \end{pmatrix} = \begin{pmatrix} -4 & 10 \\ 6 & 2 \end{pmatrix} \]

### Step 6: Solve for \( X \)
To find \( X \), we divide \( 2X \) by 2:
\[ X = \frac{1}{2} \begin{pmatrix} -4 & 10 \\ 6 & 2 \end{pmatrix} = \begin{pmatrix} -2 & 5 \\ 3 & 1 \end{pmatrix} \]

### Final Answer
Thus, the matrix \( X \) is:
\[ X = \begin{pmatrix} -2 & 5 \\ 3 & 1 \end{pmatrix} \]