Definition: Let T be a linear operator on a vector space V. A nonzero vector v ∈ V is called an eigenvector of T if there exists a scalar λ such that T(v) = λv. The scalar λ is called the eigenvalue corresponding to the eigenvector v.
Question 1): When does eigenvector $v \in V$ have to be nonzero? And why?
I also come upon this: Theorem: Let $A \in M_{n \times n}(F)$, then a scalar $\lambda$ is an eigenvalue of A if and only if $det(A-\lambda I_n)=0$
Question 2: Does that mean the linear map $(A-\lambda I_n)$ is always noninvertible?
If you allow an eigenvector to be $\mathbf{0}$, then suddenly every scalar $\lambda$ is an eigenvalue. This is because $A\mathbf{0} = \lambda \mathbf{0} = \mathbf{0}$ now holds for every $\lambda$. This is not useful.
Now $A-\lambda I$ is no longer one-to-one if there is non-zero $\mathbf{v}$ with $A\mathbf{v} = \lambda \mathbf{v}$, because both $\mathbf{0}$ and $\mathbf{v}$ are in the kernel. Hence $A-\lambda I$ must be singular/non-invertible.