If `2X-Y=[(3,-3,0),(3,3,2)]` and `2Y+X=[(4,1,5),(-1,4,-4)]`, then `X=`…………………..

Answer

To solve the given problem, we need to find the matrix \( X \) given the equations:

1. \( 2X - Y = \begin{pmatrix} 3 & -3 & 0 \\ 3 & 3 & 2 \end{pmatrix} \)
2. \( 2Y + X = \begin{pmatrix} 4 & 1 & 5 \\ -1 & 4 & -4 \end{pmatrix} \)

Let's denote the matrices as follows:
- Let \( A = \begin{pmatrix} 3 & -3 & 0 \\ 3 & 3 & 2 \end{pmatrix} \)
- Let \( B = \begin{pmatrix} 4 & 1 & 5 \\ -1 & 4 & -4 \end{pmatrix} \)

### Step 1: Rewrite the equations

From the first equation, we can express \( Y \) in terms of \( X \):
\[ Y = 2X - A \]

From the second equation, we can express \( Y \) in terms of \( X \):
\[ Y = \frac{1}{2}(B - X) \]

### Step 2: Set the equations for \( Y \) equal to each other

Now we have two expressions for \( Y \):
\[ 2X - A = \frac{1}{2}(B - X) \]

### Step 3: Clear the fraction

To eliminate the fraction, multiply the entire equation by 2:
\[ 2(2X - A) = B - X \]
This simplifies to:
\[ 4X - 2A = B - X \]

### Step 4: Rearrange the equation

Now, rearranging gives:
\[ 4X + X = B + 2A \]
\[ 5X = B + 2A \]

### Step 5: Substitute the values of \( A \) and \( B \)

Now substitute \( A \) and \( B \):
\[ 5X = \begin{pmatrix} 4 & 1 & 5 \\ -1 & 4 & -4 \end{pmatrix} + 2 \begin{pmatrix} 3 & -3 & 0 \\ 3 & 3 & 2 \end{pmatrix} \]

Calculating \( 2A \):
\[ 2A = \begin{pmatrix} 6 & -6 & 0 \\ 6 & 6 & 4 \end{pmatrix} \]

### Step 6: Add the matrices

Now add \( B \) and \( 2A \):
\[ B + 2A = \begin{pmatrix} 4 & 1 & 5 \\ -1 & 4 & -4 \end{pmatrix} + \begin{pmatrix} 6 & -6 & 0 \\ 6 & 6 & 4 \end{pmatrix} \]
\[ = \begin{pmatrix} 4 + 6 & 1 - 6 & 5 + 0 \\ -1 + 6 & 4 + 6 & -4 + 4 \end{pmatrix} \]
\[ = \begin{pmatrix} 10 & -5 & 5 \\ 5 & 10 & 0 \end{pmatrix} \]

### Step 7: Solve for \( X \)

Now substitute back to find \( X \):
\[ 5X = \begin{pmatrix} 10 & -5 & 5 \\ 5 & 10 & 0 \end{pmatrix} \]
To find \( X \), divide both sides by 5:
\[ X = \frac{1}{5} \begin{pmatrix} 10 & -5 & 5 \\ 5 & 10 & 0 \end{pmatrix} \]
\[ = \begin{pmatrix} 2 & -1 & 1 \\ 1 & 2 & 0 \end{pmatrix} \]

### Final Answer
Thus, the matrix \( X \) is:
\[ X = \begin{pmatrix} 2 & -1 & 1 \\ 1 & 2 & 0 \end{pmatrix} \]