Say A $\in M^{nxn}(F)$. Prove that the following propositions are equivalent:
i) $\lambda \in F$ is an eigenvalue of A
ii) The system $(A-\lambda I_n)X=0$ has nontrivial solutions
iii) $\exists v \in F^n $- {$0$} such that $Av=\lambda v$
iv)$\lambda \in F$ is a root of the characteristic polynomial $p_A(\lambda) =det(\lambda I_n -A) $
Let's prove first i $\leftrightarrow$ iii
If $\lambda \in F$ is an eigenvalue of A $\leftrightarrow \exists v \in F^n - ${$0$} such that $Av = \lambda v$ for the definition of an eigenvalue
Then: ii $\leftrightarrow$ iii
If the system $(A-\lambda I_n)X=0$ has nontrivial solutions $\rightarrow$ $\exists v \in F^n $- {$0$} such that $(A-\lambda I_n)v =0 $ i.e $Av=\lambda v$
And at last iv $\leftrightarrow$ iii
iv) If $\lambda \in F$ is a root of the characteristic polynomial $p_A(\lambda)=det(\lambda I_n -A) \leftrightarrow det(\lambda I_n -A) =0 \leftrightarrow A-\lambda I_n $ is non-singular $\leftrightarrow \exists v \in F^n - {$0$}$ such that $ Av = \lambda v$
Can I get feedback if this proof is complete?