What do the eigenvalues of a matrix tell us about the original matrix? I have a problem of...

Let $A$ be a 2x2 matrix such that it is not invertible and 2 is an
  eigenvalue of $A$.   
a) Find all eigenvalues of $A+I$. 
b) Prove or disprove A+I is invertible.

Since it's not invertible, it has an eigenvalue of 0. So I can think of a matrix easily such as the one below with eigenvalues of 2 and 0...
$\begin{bmatrix}
0 & a \\
0 & 2
\end{bmatrix}$
Where $a$ is just some unknown. However, I'm assuming there are many matrices that have eigenvalues of 2 and 0 for a 2x2. So I am having trouble even seeing what the eigenvalues will even tell me about the original matrix. 
Do eigenvalues tell you anything about the original structure of the matrix?  
Also I haven't learned about eigenvectors yet in class.

You don't have to find the original matrix to answer the question. As you said the eigenvalues must be $\lambda =0,2$. Claim, $\mu =1,3$ are the eigenvalues of $A+I$. Lets check, Let $x$ be an eigenvector of $A$ corresponding to the eigenvalue 2, then $$(A+I)x=Ax+x=2x+x=3x.$$Thus, $\mu=3$ is an eigenvalue for $A+I$. Similarly, we can conclude that $\mu=1$ is the other eigenvalue. Which means $A+I$ is invertible. 
