Invertibiliity of a square matrix and characteristic polynomial

Here is a nice article on a subject,

https://math.stackexchange.com/questions/166219/is-a-matrix-with-characteristic-polynomial-t2-1-invertible

In fact, as shown in the following photo, one can prove the statement next to the highlighted statement easily.

A proof goes as follows,

It is clear that $\lambda$ is an eigenvalue of $A$ iff $Det(A-\lambda I)=0$. As a consequence, $\lambda=0$ is an eigenvalue of $A$ iff $Det(A-0\cdot I)=|A|=0$. Hence, $|A|\neq 0$ iff $\lambda=0$ is not an eigenvalue of $A$ iff the characteristic polynomial of $A$ does not have zero at zero.