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.

The Given matrix has two distinct eigenvalues $0$ and $2$. Therefore it will have $2$ linearly independent eigenvectors and will be diagonalizable. That is, it will be similar to 
$$
    \begin{bmatrix}
    2 & 0 \\
    0 & 0 \\
    \end{bmatrix}
$$
So we can right $A+I$ as
\begin{bmatrix}
    3 & 0 \\
    0 & 1 \\
    \end{bmatrix}
which is a invertible matrix with eigenvalues 3 and 1.
