## Example: Solving a 3-by-3 Homogeneous System by Row Reduction

Solve the homogeneous system:
```
 x + 2y + 3z = 0
2x + 5y + 3z = 0
 x + 8z = 0
```

**Step 1:** Write augmented matrix (right column is always 0 for homogeneous systems)
```
[ 1  2  3 | 0]
[ 2  5  3 | 0]
[ 1  0  8 | 0]
```

**Step 2:** R₂ → R₂ - 2R₁, R₃ → R₃ - R₁
```
[ 1  2  3 | 0]
[ 0  1 -3 | 0]
[ 0 -2  5 | 0]
```

**Step 3:** R₃ → R₃ + 2R₂
```
[ 1  2  3 | 0]
[ 0  1 -3 | 0]
[ 0  0 -1 | 0]
```

**Step 4:** R₃ → -R₃
```
[ 1  2  3 | 0]
[ 0  1 -3 | 0]
[ 0  0  1 | 0]
```

**Step 5:** Back-substitute: z = 0, y = 3(0) = 0, x = -2(0) - 3(0) = 0

**Solution:** The only solution is the trivial solution x = 0, y = 0, z = 0.

This homogeneous system has only the trivial solution because the coefficient matrix has full rank (3 pivots). When a homogeneous system has free variables, nontrivial solutions exist.