Is it possible to solve a linear matrix system $A x = b$ using the Newton-Raphson method? If yes, how can this be done? More special, how is the derivative build?
Yes you can do this, and it will converge in one iteration regardless of the starting value.
This is because each step of Newton's method involves solving a linear system with the Jacobian of the nonlinear function. In this case the Jacobian just equals $A$.
In other words: this is a little circular because it requires you to solve the system $Ax=b$ in the first place, which doesn't help.