Matrices and Complex Numbers Given this set:
$$
S=\left\{\begin{bmatrix}a&-b\\b&a\end{bmatrix}\middle|\,a,b\in\Bbb R\right\}
$$
Part I:
Why is this set equivalent to the set of all complex numbers a+bi (when both are under multiplication?)
There is one matrix that corresponds to a specific complex number. Can this example be found and how can it be demonstrated to give equivalent answers?
Part II:
What is a formula for the multiplicative inverse of the matrix shown in the set, using knowledge on inverses of complex numbers?

Looking in the comments since I've posted, the following answer appears to be expanding on Jyrki's idea.
There exists a homomorphism $\phi:S \rightarrow \mathbb{C}$ defined as follows:
$$\begin{bmatrix}
       a & -b \\[0.3em]
       b & a  \\[0.3em]
\end{bmatrix} \mapsto (a + bi)$$ 
Of course, you will want to prove that this is indeed a homomorphism by checking the following conditions:


*

*$\phi$ maps the multiplicative identity in $S$ to the multiplicative identity in $\mathbb{C}$.

*$\phi(xy) = \phi(x)\phi(y)$ for any $x, y \in S$.

*$\phi(x + y) = \phi(x) + \phi(y)$ for any $x, y \in S$.


Once you have done this, then show that $Im(\phi) = \mathbb{C}$, and $\ker(\phi) = \{0\}$, where $0$ is the additive identity in $S$.  From here, you can apply the isomorphism theorem to show that $S$ is isomorphic to $\mathbb{C}$.
