Answer
$\lambda=-1$
Work Step by Step
If A is an nxn matrix, a scalar $\lambda$ is called an eigenvalue of A is there is a non-zero vector $v$ such that $Ax=\lambda x$. This vector is called the eigenvector of A corresponding to $\lambda$.
Hence here, we compute:
$Av=\begin{bmatrix}
1 &2 \\
2 & 1
\end{bmatrix}\cdot\begin{bmatrix}
3 \\
-3
\end{bmatrix}=\begin{bmatrix}
-3\\
3
\end{bmatrix}=-1\begin{bmatrix}
3 \\
-3
\end{bmatrix}=-1v$
Thus we can see that $\lambda=-1$
