If an n x n matix A is not invertible, then is 0 an eigenvalue of A? I know that if $0$ is an eigenvalue of a matrix $A$, then $A$ is not invertible, but is the opposite also true? If the matrix isn't invertible, is the eigenvalue always $0$?

The condition of non-invertibility amounts to $\det(A)=0$ which in turn is equivalent to that the rank of $A$ is not maximal. But that means that you can find a non-zero column $X$ such that
$$
A\cdot X=0.
$$
Thinking of $X$ as a vector, the displayed formula says that $X$ is an eigenvector for the eigenvalue $0$.
