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To solve the given problem, we need to find the solutions of the homogeneous system \( A\mathbf{x} = 0 \), where the matrix \( A \) is provided:

\[
A = \begin{bmatrix} 
8 & 1 & -8 \\
16 & 2 & -16 
\end{bmatrix}
\]

We will proceed with the following steps:

### 1. Row Reduce the Matrix:
The goal is to row reduce the matrix to its row echelon form. We can perform elementary row operations to simplify it.

\[
\text{Initial Matrix:} \quad \begin{bmatrix} 
8 & 1 & -8 \\
16 & 2 & -16 
\end{bmatrix}
\]

First, we can replace row 2 (\(R_2\)) by \( R_2 - 2R_1 \):

\[
R_2 \rightarrow R_2 - 2R_1 = \begin{bmatrix} 
16 & 2 & -16 
\end{bmatrix} - 2 \begin{bmatrix} 
8 & 1 & -8 
\end{bmatrix} = \begin{bmatrix} 
0 & 0 & 0 
\end{bmatrix}
\]

The row-reduced matrix is now:

\[
\begin{bmatrix} 
8 & 1 & -8 \\
0 & 0 & 0 
\end{bmatrix}
\]

### 2. Analyze the Row-Reduced System:
This gives us the following system of linear equations:

\[
8x_1 + x_2 - 8x_3 = 0
\]
\[
0 = 0
\]

The second row is trivial, so we focus on the first equation:

\[
8x_1 + x_2 - 8x_3 = 0
\]

Solving for \( x_1 \):

\[
x_1 = \frac{-1}{8}x_2 + x_3
\]

### 3. Identify Free Variables:
In this system, \( x_1 \) is expressed in terms of \( x_2 \) and \( x_3 \). Therefore, \( x_2 \) and \( x_3 \) are the free variables.

### Conclusion:
The free variables are \( x_2 \) and \( x_3 \), meaning the correct answer is:

- \( x_2 \)
- \( x_3 \)

Would you like more details on any part of this process? Here are some related questions for further understanding:

1. What are the steps of Gaussian elimination in more detail?
2. How do you recognize free variables in a system of linear equations?
3. Can you explain why the second row being zero leads to infinite solutions?
4. What does the row-reduced form of a matrix represent in terms of solution sets?
5. How does this solution change if the matrix had more rows or variables?

**Tip**: When solving homogeneous systems like this, always look for rows of zeros, as they indicate free variables and imply infinite solutions.

Find all the solutions of Ax = 0, where A is row equivalent to the given matrix. The free variable(s) is/are (select all that apply): x1, x2, x3.

Explore how to find solutions for homogeneous systems using Gaussian elimination. This problem uses a matrix to solve for free variables and demonstrates the reduction to row echelon form, suitable for university-level students studying linear algebra.

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