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Let's solve the problem step by step.

### Given:
Matrix \( A \) and \( B \) are defined as:
\[
A = \begin{pmatrix} 1 & -\frac{1}{2}x \\ -2y & 3 \end{pmatrix}
\]
\[
B = \begin{pmatrix} x + y & x \\ y & x - y \end{pmatrix}
\]
It is also given that the transpose of \( A \), denoted by \( A^T \), equals \( B \).

### Step 1: Transpose of Matrix \( A \)
The transpose of matrix \( A \) is obtained by swapping its rows with columns:
\[
A^T = \begin{pmatrix} 1 & -2y \\ -\frac{1}{2}x & 3 \end{pmatrix}
\]

### Step 2: Equating \( A^T \) and \( B \)
We are given that \( A^T = B \). Therefore, equating corresponding elements from \( A^T \) and \( B \), we get:
\[
\begin{pmatrix} 1 & -2y \\ -\frac{1}{2}x & 3 \end{pmatrix} = \begin{pmatrix} x + y & x \\ y & x - y \end{pmatrix}
\]

This gives us the following system of equations:
1. \( 1 = x + y \)
2. \( -2y = x \)
3. \( -\frac{1}{2}x = y \)
4. \( 3 = x - y \)

### Step 3: Solving the System of Equations
We will solve these equations to find the values of \( x \) and \( y \).

- From equation (3): \( y = -\frac{1}{2}x \).
- Substitute this into equation (1):
  \[
  1 = x + \left(-\frac{1}{2}x\right) \implies 1 = \frac{1}{2}x \implies x = 2
  \]
- Now, substitute \( x = 2 \) back into \( y = -\frac{1}{2}x \):
  \[
  y = -\frac{1}{2} \times 2 = -1
  \]
  
### Step 4: Calculate \( x + 2y \)
Finally, calculate \( x + 2y \):
\[
x + 2y = 2 + 2(-1) = 2 - 2 = 0
\]

### Answer:
The value of \( x + 2y \) is \( \boxed{0} \).

Would you like any further explanation or have any other questions?

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Here are some additional related questions:
1. What is the determinant of the matrix \( A \)?
2. How would you find the inverse of matrix \( A \) if it exists?
3. What are the eigenvalues of matrix \( A \)?
4. How does the rank of a matrix relate to its determinant?
5. Can you find the trace of matrix \( A \) and \( B \)?

**Tip:** When dealing with matrices, always check if they are square when asked to find properties like determinants, inverses, or eigenvalues, as these are only defined for square matrices.

Diketahui matriks A = \begin{pmatrix} 1 & -\frac{1}{2}x \\ -2y & 3 \end{pmatrix}, dan B = \begin{pmatrix} x + y & x \\ y & x - y \end{pmatrix}, dengan A^T = B. Nilai x + 2y adalah ...

Learn how to solve a matrix transpose equality problem where A^T = B. The problem involves transposing matrices, setting up systems of equations, and solving for x and y. Suitable for undergraduate mathematics students.

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