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.

Proof

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Sources

• 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.2$: Functions on vectors: $\S 2.2.5$: Determinants: Lemma $2.2.1$