Answer
$\lambda=3$
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:
$Av=\begin{bmatrix}
0 &3 \\
3 & 0
\end{bmatrix}\cdot\begin{bmatrix}
1 \\
1
\end{bmatrix}=\begin{bmatrix}
3\\
3
\end{bmatrix}=3\begin{bmatrix}
1 \\
1
\end{bmatrix}=3v$
Thus we can see that $\lambda=3$