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

Answer

To find the matrix \( X \) such that \( A + X = 2B + C \), we will follow these steps:

### Step 1: Write down the matrices
We have the following matrices:
\[ A = \begin{pmatrix} 2 & -1 \\ 2 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} -3 & 2 \\ 4 & 0 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} \]

### Step 2: Calculate \( 2B \)
First, we need to calculate \( 2B \):
\[ 2B = 2 \times \begin{pmatrix} -3 & 2 \\ 4 & 0 \end{pmatrix} = \begin{pmatrix} 2 \times -3 & 2 \times 2 \\ 2 \times 4 & 2 \times 0 \end{pmatrix} = \begin{pmatrix} -6 & 4 \\ 8 & 0 \end{pmatrix} \]

### Step 3: Add \( 2B \) and \( C \)
Next, we will add \( 2B \) and \( C \):
\[ 2B + C = \begin{pmatrix} -6 & 4 \\ 8 & 0 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} -6 + 1 & 4 + 0 \\ 8 + 0 & 0 + 2 \end{pmatrix} = \begin{pmatrix} -5 & 4 \\ 8 & 2 \end{pmatrix} \]

### Step 4: Set up the equation for \( X \)
Now we can set up the equation:
\[ A + X = 2B + C \]
Substituting the values we have:
\[ \begin{pmatrix} 2 & -1 \\ 2 & 0 \end{pmatrix} + X = \begin{pmatrix} -5 & 4 \\ 8 & 2 \end{pmatrix} \]

### Step 5: Solve for \( X \)
To find \( X \), we rearrange the equation:
\[ X = \begin{pmatrix} -5 & 4 \\ 8 & 2 \end{pmatrix} - \begin{pmatrix} 2 & -1 \\ 2 & 0 \end{pmatrix} \]
Now we perform the subtraction:
\[ X = \begin{pmatrix} -5 - 2 & 4 - (-1) \\ 8 - 2 & 2 - 0 \end{pmatrix} = \begin{pmatrix} -7 & 5 \\ 6 & 2 \end{pmatrix} \]

### Final Answer
Thus, the matrix \( X \) is:
\[ X = \begin{pmatrix} -7 & 5 \\ 6 & 2 \end{pmatrix} \]

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