Prove that matrix W is a field

Let $\displaystyle W=\left\{ \left[ \begin{matrix} a & -b \\ b & a \end{matrix} \right] : a,b\in R \right\}$ together with the usual operations of matrix addition and matrix multiplication in $\displaystyle R^{2\times 2}$. Prove that $\displaystyle W$ is a field.

A hint was provided as follows:

Prove that $\displaystyle W$ is closed under addition and multiplication and then go through each axiom of a field. You may simply reference well-known facts about matrix multiplication, i.e. $\displaystyle (A+B)+C=A+(B+C)$ $\displaystyle \forall A,B,C\in R^{2\times 2}$.

This is probably easier than I think it is, but I'm still highly uncertain when it comes to fields and matrices.