If X + Y = `[{:(7, 0),(2,5):}]` and X - Y = `[{:(3,0),(0,3):}]` , then

Options

A. `X = [{:(5,0),(1,4):}], Y = [{:(2,0),(0,1):}]`
B. `X = [{:(5,0),(1,4):}], Y = [{:(2,0),(1,1):}]`
C. `X = [{:(2,1),(1,0):}], Y = [{:(1,2),(4,3):}]`
D. `X = [{:(5,4),(1,0):}], Y = [{:(2,3),(4,5):}]`

Answer

To solve the given problem, we have two matrix equations:

1. \( X + Y = \begin{pmatrix} 7 & 0 \\ 2 & 5 \end{pmatrix} \) (Equation 1)
2. \( X - Y = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \) (Equation 2)

We need to find the matrices \( X \) and \( Y \).

### Step 1: Add the two equations

To eliminate \( Y \), we can add Equation 1 and Equation 2:

\[ (X + Y) + (X - Y) = \begin{pmatrix} 7 & 0 \\ 2 & 5 \end{pmatrix} + \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \]

This simplifies to:

\[ 2X = \begin{pmatrix} 7 + 3 & 0 + 0 \\ 2 + 0 & 5 + 3 \end{pmatrix} = \begin{pmatrix} 10 & 0 \\ 2 & 8 \end{pmatrix} \]

### Step 2: Solve for \( X \)

Now, divide both sides by 2 to find \( X \):

\[ X = \frac{1}{2} \begin{pmatrix} 10 & 0 \\ 2 & 8 \end{pmatrix} = \begin{pmatrix} 5 & 0 \\ 1 & 4 \end{pmatrix} \]

### Step 3: Substitute \( X \) back to find \( Y \)

Now that we have \( X \), we can substitute it back into either Equation 1 or Equation 2 to find \( Y \). Let's use Equation 1:

\[ X + Y = \begin{pmatrix} 7 & 0 \\ 2 & 5 \end{pmatrix} \]

Substituting \( X \):

\[ \begin{pmatrix} 5 & 0 \\ 1 & 4 \end{pmatrix} + Y = \begin{pmatrix} 7 & 0 \\ 2 & 5 \end{pmatrix} \]

### Step 4: Solve for \( Y \)

Now, isolate \( Y \):

\[ Y = \begin{pmatrix} 7 & 0 \\ 2 & 5 \end{pmatrix} - \begin{pmatrix} 5 & 0 \\ 1 & 4 \end{pmatrix} \]

Subtracting the matrices element-wise gives:

\[ Y = \begin{pmatrix} 7 - 5 & 0 - 0 \\ 2 - 1 & 5 - 4 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix} \]

### Final Result

Thus, the matrices \( X \) and \( Y \) are:

\[ X = \begin{pmatrix} 5 & 0 \\ 1 & 4 \end{pmatrix}, \quad Y = \begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix} \]