**Theorem:** Every homogeneous linear system $Ax = 0$ is **consistent**.

**Proof:** The zero vector $x = \mathbf{0}$ always satisfies every equation because:
$$a_{i1} \cdot 0 + a_{i2} \cdot 0 + \cdots + a_{in} \cdot 0 = 0$$

for every row $i$.

This means a homogeneous system always has at least one solution (the trivial solution $x = 0$).

**Important consequence:** For homogeneous systems, the only question is whether the solution is unique (only $x = 0$) or whether there are infinitely many solutions (nontrivial solutions exist).