Eigenvalues and eigenspaces/Characterization of an automorphism by eigenvalues/Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space, and let ${}\varphi \in \operatorname {End} _{}{\left(V\right)}$. Show that the following statements are equivalent:

1. The linear mapping ${}\varphi$ is an isomorphism.
2. ${}0$ is not an eigenvalue of ${}\varphi$.
3. The constant term of the characteristic polynomial ${}\chi _{\varphi }$ is ${}\neq 0$.

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