Matrix is Non-Invertible iff Product with Non-Zero Vector is Zero

Theorem
Let $\mathbf A$ be a square matrix of order $n$.

Then $\mathbf A$ is non-invertible if there exists a vector $\mathbf v$ of $n$ such that:
 * $\mathbf v \ne \mathbf 0$
 * $\mathbf A \mathbf v = \mathbf 0$

where $\mathbf 0$ is the zero vector.