Characteristic polynomial/Insert and direct determinant/Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}M}$ denote an ${\displaystyle {}n\times n}$-matrix over ${\displaystyle {}K}$. Show that for every ${\displaystyle {}\lambda \in K}$, the relation

${\displaystyle {}\chi _{M}(\lambda )=\det {\left(\lambda E_{n}-M\right)}\,}$

holds.