this is a two part question...

1) i need to prove that the set F of all the 2x2 real matrices of the form

$\displaystyle A = \begin{bmatrix}

a & b\\

-b & a

\end{bmatrix}$

is a field with operations matrix addition and matrix multiplication.

2) Prove that F is isomorphic to $\displaystyle \mathbb{C}$

I was gonna define

$\displaystyle \varphi : F \to \mathbb{C}$ by $\displaystyle \varphi (A)=a+ib$...

i think this is obvious but not sure where to go here.. because if we are talking about real matrices, then how can it be isomorphic to a complex numbers?