The given exercise is, for all $\theta$ in $\mathbb{R}$, show that the matrix always has an eigenvector in $\mathbb{R^2}$ $$
A = \begin{pmatrix}
\cos(\theta) & \sin(\theta) \\
\sin(\theta) & -\cos(\theta)
\end{pmatrix}
$$

The use of determinants isn't allowed, it is just allowed that I should use de definition of eigenvector and eigenvalue, and the fact that the eigenvectors are linearly independent if they are distinct (if needed). 

I don't fully understand the problem because I can't see how is possible that this matrix can give me a multiple scalar of some vector.