Proof of Theorem SMZE

We have the following equivalences: \begin{align*} \text{$A$ is singular}&\iff\text{there exists $\vect{x}\neq\zerovector$, $A\vect{x}=\zerovector$}&&\knowl{./knowls/definition.NM.knowl}{\text{Definition NM}}\\ &\iff\text{there exists $\vect{x}\neq\zerovector$, $A\vect{x}=0\vect{x}$}&&\knowl{./knowls/theorem.ZSSM.knowl}{\text{Theorem ZSSM}}\\ &\iff\text{$\lambda=0$ is an eigenvalue of $A$}&&\knowl{./knowls/definition.EEM.knowl}{\text{Definition EEM}} \end{align*}