Get rid of that
//MatrixForm
and then you can successfully
In[1]:= ...
Aug1 = Normal[ArrayFlatten[{{A, B}}]];
RowReduce[Aug1]
Out[9]= {{1, 0, 0, 0, 33/20}, {0, 1, 0, 0, -(4/5)}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 5/4}}
//MatrixForm might make an expression cute to look at, but you cannot then do any calculations on your cute result.
If all Mathematica functions would look at anything that was a MatrixForm or any of the other *Form, strip that off and then do the calculation as if it were never present then this would avoid repeated confusion and questions like this. Users seem to be like that, every new pretty printing feature starts getting used and then the questions begin asking why they then can't compute using that pretty result.
The same could be said of much of the desktop publishing functionality. Unfortunately, the things users can do with that seems like will make it far harder for Mathematica to make sense of and correctly remove to enable sensible calculations.