Suppose we have the following square matrix of order 3:

$$A = \begin{pmatrix}
1 & 3 & 3\\ 
-3 & -5 & -3 \\ 
3 & 3 & 1
\end{pmatrix}$$

How can I prove that $\lambda=-2$ is eigenvalue without having to calculate the characteristic polynomial and diagonalization?

I know the definition of eigenvalue and I know how to diagonalize. But, I wouldn't know how to do that.