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?

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.