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To solve this problem, we need to understand that the set \( H = \{(x, y, z) | 6x + 6y - 9z = 0\} \) represents a subspace of \( \mathbb{R}^3 \), and we are looking for a matrix \( A \) such that \( H = N(A) \), where \( N(A) \) denotes the null space of matrix \( A \).

### Step 1: Analyzing the Equation
The subspace \( H \) is defined by the equation \( 6x + 6y - 9z = 0 \), which can be simplified by dividing through by 3:
\[
2x + 2y - 3z = 0
\]
This is the equation of a plane in \( \mathbb{R}^3 \).

### Step 2: Matrix Representation for Null Space
We are tasked with finding a matrix \( A \) such that the null space of \( A \) corresponds to the solution set of the equation \( 2x + 2y - 3z = 0 \). The null space is the set of vectors \( (x, y, z) \) that satisfy this equation.

This can be written in matrix form as:
\[
\begin{pmatrix}
2 & 2 & -3
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix} = 0
\]
So the matrix \( A \) must have a row that represents this linear combination. In order for the null space to be described correctly, the matrix can be:
\[
A = \begin{pmatrix}
2 & 2 & -3
\end{pmatrix}
\]

### Step 3: Checking the Given Options
Now, let's check the options:

- **Option a**:
  \[
  A = \begin{pmatrix}
  4 & 4 & -3 \\
  0 & 0 & 0 \\
  6 & 6 & 9
  \end{pmatrix}
  \]
  The first row is not a scalar multiple of \( (2, 2, -3) \), so this is incorrect.

- **Option b**:
  \[
  A = \begin{pmatrix}
  2 & -3 & 2 \\
  6 & 9 & 4 \\
  -2 & -3 & 2
  \end{pmatrix}
  \]
  This doesn't match the structure we are looking for.

- **Option c**:
  \[
  A = \begin{pmatrix}
  2 & -1 & 1 \\
  0 & 0 & 0 \\
  2 & -1 & -1
  \end{pmatrix}
  \]
  This is not consistent with the desired form either.

- **Option d**:
  \[
  A = \begin{pmatrix}
  4 & 4 & -3 \\
  0 & 0 & 0 \\
  0 & 0 & 0
  \end{pmatrix}
  \]
  The first row is a multiple of \( (2, 2, -3) \), so this matrix is consistent with the null space equation.

### Conclusion:
The correct answer is **Option d**.

Would you like a detailed explanation of null spaces or matrices?

Here are some related questions:
1. How is the null space of a matrix defined?
2. What is the geometrical interpretation of the null space?
3. Can a matrix have a trivial null space?
4. How does row reduction help in finding the null space?
5. What is the rank-nullity theorem?

**Tip**: To find the null space, always convert the equation into matrix form and ensure the solution set of the system matches the null space.

Sea H = {(x, y, z) | 6x + 6y - 9z = 0}. Una matriz A tal que H = N_A es...

Learn how to find a matrix A such that its null space corresponds to the subspace defined by the equation 6x + 6y - 9z = 0. This problem involves concepts from linear algebra, including null spaces, matrix representation, and subspaces. Ideal for undergraduate students studying linear algebra.

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