If `[(x+y,y),(2x,x-y)][(2),(-1)]=[3/2]` then x.y is equal to :

Options

A. `-5`
B. 5
C. 4
D. 6

Answer

To solve the given problem step by step, we start with the equation involving matrices:

Given:
\[ \begin{pmatrix} x+y & y \\ 2x & x-y \end{pmatrix} \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} \frac{3}{2} \\ 2 \end{pmatrix} \]

### Step 1: Perform Matrix Multiplication
We will multiply the first matrix by the second matrix.

The first element of the resulting matrix is calculated as:
\[ (x+y) \cdot 2 + y \cdot (-1) = 2(x+y) - y \]

The second element of the resulting matrix is calculated as:
\[ 2x \cdot 2 + (x-y) \cdot (-1) = 4x - (x - y) = 4x - x + y = 3x + y \]

So, we have:
\[ \begin{pmatrix} 2(x+y) - y \\ 3x + y \end{pmatrix} = \begin{pmatrix} \frac{3}{2} \\ 2 \end{pmatrix} \]

### Step 2: Set Up the Equations
From the matrix multiplication, we can set up the following equations:
1. \( 2(x+y) - y = \frac{3}{2} \)
2. \( 3x + y = 2 \)

### Step 3: Simplify the First Equation
Let's simplify the first equation:
\[ 2(x+y) - y = \frac{3}{2} \]
Expanding it gives:
\[ 2x + 2y - y = \frac{3}{2} \]
This simplifies to:
\[ 2x + y = \frac{3}{2} \quad \text{(Equation 1)} \]

### Step 4: Use the Second Equation
The second equation is:
\[ 3x + y = 2 \quad \text{(Equation 2)} \]

### Step 5: Solve the System of Equations
Now we have a system of equations:
1. \( 2x + y = \frac{3}{2} \)
2. \( 3x + y = 2 \)

We can eliminate \(y\) by subtracting Equation 1 from Equation 2:
\[ (3x + y) - (2x + y) = 2 - \frac{3}{2} \]
This simplifies to:
\[ 3x - 2x = 2 - \frac{3}{2} \]
\[ x = 2 - \frac{3}{2} = \frac{4}{2} - \frac{3}{2} = \frac{1}{2} \]

### Step 6: Substitute Back to Find \(y\)
Now substitute \(x = \frac{1}{2}\) back into either equation to find \(y\). We will use Equation 1:
\[ 2\left(\frac{1}{2}\right) + y = \frac{3}{2} \]
This simplifies to:
\[ 1 + y = \frac{3}{2} \]
Thus:
\[ y = \frac{3}{2} - 1 = \frac{1}{2} \]

### Step 7: Calculate \(x \cdot y\)
Now we have \(x = \frac{1}{2}\) and \(y = \frac{1}{2}\). Therefore:
\[ x \cdot y = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \]

### Final Answer
Thus, the value of \(x \cdot y\) is:
\[ \boxed{\frac{1}{4}} \]