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The system of equations given in matrix form is:

\[
\begin{pmatrix}
2 & 1 & 3 \\
1 & 0 & 5 \\
0 & 1 & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
=
\begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix}
\]

We will solve this by row-reducing the augmented matrix:

\[
\left[\begin{array}{ccc|c}
2 & 1 & 3 & 0 \\
1 & 0 & 5 & 0 \\
0 & 1 & 1 & 0
\end{array}\right]
\]

Let's proceed with the row-reduction steps to find the solution.The row-reduced augmented matrix is:

\[
\left[\begin{array}{ccc|c}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0
\end{array}\right]
\]

This corresponds to the system of equations:

\[
x_1 = 0, \quad x_2 = 0, \quad x_3 = 0
\]

Thus, the solution to the system is \( x_1 = 0 \), \( x_2 = 0 \), and \( x_3 = 0 \), which is the trivial solution.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the significance of the trivial solution in homogeneous systems of equations?
2. How can row-reduction help determine if a system has infinitely many solutions?
3. How do you interpret a system that has no solutions after row reduction?
4. What are the differences between the row echelon form and the reduced row echelon form?
5. How can matrix row operations affect the rank of a matrix?

**Tip**: When performing row reductions, ensure that you focus on creating leading ones and zeros below and above them, which helps to identify the solution structure quickly.

Solve the system of equations by row-reducing the augmented matrix: 
\(
\begin{pmatrix}
2 & 1 & 3 \\
1 & 0 & 5 \\
0 & 1 & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
=
\begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix}
\).

Learn how to solve a homogeneous system of equations using row-reducing the augmented matrix. This example walks through the process of Gaussian elimination and matrix row operations, suitable for Linear Algebra students.

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