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We are given the transformation \( T(x, y, z) \) as follows:

\[
T(x, y, z) = \begin{bmatrix} x + z \\ 2y \end{bmatrix}
\]

### Problem 1: Find a matrix \( A \) such that \( T(x) = A x \).

To represent this transformation as a matrix, we will express \( T(x, y, z) \) as a matrix multiplication. Assume \( x \), \( y \), and \( z \) are represented as a column vector:

\[
\begin{bmatrix} x \\ y \\ z \end{bmatrix}
\]

We are looking for a matrix \( A \) such that:

\[
T(x, y, z) = A \begin{bmatrix} x \\ y \\ z \end{bmatrix}
\]

### Step-by-Step Solution:

#### Step 1: Write the transformation in terms of components
The transformation is given as:

\[
T(x, y, z) = \begin{bmatrix} x + z \\ 2y \end{bmatrix}
\]

We want to find a matrix \( A \) that transforms the vector \( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \) into \( \begin{bmatrix} x + z \\ 2y \end{bmatrix} \).

#### Step 2: Analyze the first component
The first component of the output vector is \( x + z \). This suggests that the first row of matrix \( A \) should take the \( x \)-coordinate and add the \( z \)-coordinate, while ignoring the \( y \)-coordinate. Therefore, the first row of \( A \) will be:

\[
\begin{bmatrix} 1 & 0 & 1 \end{bmatrix}
\]

This row represents the transformation of the input vector to produce the first component \( x + z \).

#### Step 3: Analyze the second component
The second component of the output vector is \( 2y \). This suggests that the second row of matrix \( A \) should take the \( y \)-coordinate, multiply it by 2, and ignore the \( x \)- and \( z \)-coordinates. Therefore, the second row of \( A \) will be:

\[
\begin{bmatrix} 0 & 2 & 0 \end{bmatrix}
\]

This row represents the transformation of the input vector to produce the second component \( 2y \).

#### Step 4: Construct the matrix \( A \)
Now, we combine the two rows we derived into a single matrix. The matrix \( A \) is:

\[
A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \end{bmatrix}
\]

Thus, the transformation \( T(x, y, z) \) can be written as:

\[
T(x, y, z) = A \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}
\]

Verifying:

\[
A \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \cdot x + 0 \cdot y + 1 \cdot z \\ 0 \cdot x + 2 \cdot y + 0 \cdot z \end{bmatrix} = \begin{bmatrix} x + z \\ 2y \end{bmatrix}
\]

This confirms that the matrix \( A \) is correct.

---

Would you like more details or have any further questions? Here are 5 related questions to consider:

1. How would the matrix change if the transformation was \( T(x, y, z) = \begin{bmatrix} x + 2z \\ 3y \end{bmatrix} \)?
2. How do you find the inverse of matrix \( A \), assuming it exists?
3. What are the conditions for a matrix transformation to be invertible?
4. How can you interpret matrix transformations geometrically in 3D space?
5. How do you compute the determinant of matrix \( A \), and what does it represent?

**Tip:** Understanding matrix transformations helps in visualizing how vectors in space are stretched, rotated, or sheared through multiplication.

Find a matrix A such that T(x) = Ax for T(x, y, z) = [x + z; 2y].

Learn how to find a matrix A for the transformation T(x, y, z) = [x + z; 2y]. This problem explores matrix multiplication and linear transformations, suitable for students in Grades 10-12 or introductory undergraduate courses.

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