  The theory of generalized Weyl algebras is used to study the $2\times 2$
reflection equation algebra $\mathcal{A}=\mathcal{A}_q(\operatorname{M}_2)$ in
the case that $q$ is not a root of unity, where the $R$-matrix used to define
$\mathcal{A}$ is the standard one of type $A$. Simple finite dimensional
$\mathcal{A}$-modules are classified, finite dimensional weight modules are
shown to be semisimple, $\operatorname{Aut}(\mathcal{A})$ is computed, and the
prime spectrum of $\mathcal{A}$ is computed along with its Zariski topology.
Finally, it is shown that $\mathcal{A}$ satisfies the Dixmier-Moeglin
equivalence.
