**Theorem:** An $n \times n$ matrix $A$ is nonsingular if and only if $Ax = 0$ has only the **trivial solution** $x = 0$.

**Proof:** If $A$ is nonsingular, $x = A^{-1}0 = 0$. Conversely, if only $x = 0$ solves $Ax = 0$, then the null space is $\{0\}$, so nullity is 0, so rank is $n$, so columns are linearly independent, so $A$ is nonsingular.