How can I get the same matrix from simpleitk? I tried to get the Direction Matrix and multiply it with the Spacing, than adding the extra row and column. Howewer the result looses the negative signs of the first two nonzero coordinates. How can I receive those as negative from simpleitk?

Nifti uses an RAS coordinate system and ITK/SimpleITK use an LPS one which is why there is a difference in the axis signs. For a detailed description of the various commonly used coordinate systems and how to convert between them see the 3D slicer documentation on coordinate systems.