How can I compute $A(v_1 + v_2)$ where $v_1$ and $v_2$ are eigenvectors of the matrix A 
If $v_1 = \begin{bmatrix}5\\3\end{bmatrix}$ and $v_2 = \begin{bmatrix}3\\1\end{bmatrix}$ are eigenvectors of a matrix $A$ corresponding to the eigenvalues $\lambda_1 = -1$ and $\lambda_2 = 4$ respectively, then
$A(v_1 + v_2)= ?$
What about $A(3v_1) = ?$

I get what I'm doing wrong. I did $v_1 + v_2 = \begin{bmatrix}5\\3\end{bmatrix} + \begin{bmatrix}3\\1\end{bmatrix} = \begin{bmatrix}8\\4\end{bmatrix}$ then I did $A(\begin{bmatrix}8\\4\end{bmatrix})= \begin{bmatrix}-1&0\\0&4\end{bmatrix}*\begin{bmatrix}8\\4\end{bmatrix} = \begin{bmatrix}-8\\16\end{bmatrix}$

$A$ is only diagonal with respect to a basis of eigenvectors, namely $v_1$ and $v_2$, not the standard basis, which is why your computation fails. Try using linearity and the definition of eigenvectors instead: 
$A(v_1+v_2)=Av_1+Av_2=\lambda_1 v_1+\lambda_2 v_2$
Similarly for $A(3v_1)$. 
