Representing the modulus and complex conjugate in matrix terms

Hi, I have the following two questions...I am not sure if I am in the right direction with the first one and I have no clue about the second one :-/

1.- Consider the set C of all matrices (with real entries) of the form

(sorry, I don't know how to code matrices! I'll separate each element with "|")

(a | -b)

(b | a)

The set C of matrices can clearly be identified with the complex numbers.

So I am a bit lost here... Why? Is it because, for example, if z=a+ib, its complex conjugate will be a-ib?

The next question is...

Continuing the previous exercise, how do the modulus and complex conjugate appear when translated into matrix terms? How does the reciprocal of the complex number z appears?

I would really appreciate it if you could give me a hint, thanks! I also tried to write the matrices using the code, but I go it wrong... I do not know if I am using an old version of the code or something