https://file.aiquickdraw.com/mathbot/file/1725868184540g8jkye4e.jpg

Let's analyze the system of linear equations based on the matrix provided.

The matrix is:

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

This corresponds to the following system of homogeneous linear equations (since it's homogeneous, the right-hand side is all zeros):

1. \( x_1 + x_4 = 0 \)
2. \( x_3 = 0 \)
3. No equation for the third row (as it is entirely zero).

Now, let's solve this:

### From the first equation:
\( x_1 + x_4 = 0 \)  
\(\Rightarrow x_1 = -x_4 \)

### From the second equation:
\( x_3 = 0 \)

### Free variables:
- \( x_2 \) and \( x_4 \) are free variables because they are not restricted by any equations.

### General solution:
We can express \( x_1, x_2, x_3, x_4 \) in terms of the free variables \( x_2 \) and \( x_4 \), using parameters \( t \) and \( s \) respectively.

Let:
- \( x_2 = t \)
- \( x_4 = s \)

Thus, the general solution is:
\[
x_1 = -s, \quad x_2 = t, \quad x_3 = 0, \quad x_4 = s
\]

In vector form, the solution is:
\[
(x_1, x_2, x_3, x_4) = (-s, t, 0, s)
\]

### Final answer:
\[
(x_1, x_2, x_3, x_4) = (-s, t, 0, s)
\]

Would you like further details or have any other questions?

Here are some related questions to deepen your understanding:

1. What is a homogeneous system of linear equations?
2. How do you determine free variables in a linear system?
3. What does it mean for a system to have an infinite number of solutions?
4. How would the solution change if the system was non-homogeneous?
5. How can matrix row reduction help solve systems of linear equations?

**Tip:** In a homogeneous system, if there are free variables, it implies there are infinitely many solutions!

Solve the homogeneous linear system corresponding to the given coefficient matrix.

Explore the solution of a homogeneous linear system with an infinite number of solutions by expressing variables in terms of parameters. This problem uses matrix algebra to analyze a linear system, and is suitable for undergraduate students.

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