Construction of characteristics polynomial

Question

Suppose we have a matrix $A$ of order $n$. Let $\lambda$ be an eigenvalue of $A$ and $y$ be a corresponding eigenvector. Then from the relation $Ay = \lambda y$ we get some relation about the coordinates of the eigenvector.

From those relations can we construct the characteristics polynomial?

Answer 1

$Ay = \lambda y \Rightarrow Ay- \lambda y=0$

$Ay- \lambda y=(A-\lambda I)y=0 \Rightarrow Ker((A- \lambda I))\neq \{0\}$

$Ker((A- \lambda I))\neq \{0\} \Rightarrow Det((A- \lambda I))=0$