1. ## Isomorphich question

this is a two part question...

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

$A = \begin{bmatrix}
a & b\\
-b & a
\end{bmatrix}$

is a field with operations matrix addition and matrix multiplication.

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

I was gonna define

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

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?

2. The link Representing complex numbers as 2&#215;2 matrices Rod Carvalho&#039;s web notebook might help with part two, but for part one I'm not sure . . .