> The difference is dataframes can also be seen like matrices. You can do row operations, row + column operations, multiply rows and columns, multiply different matrices, transpose them etc.

I think this is overblowing the similarities to matrices. Matrices have elements all of the same type, while data.frames mix numbers, characters, factors, etc. You certainly cannot transpose a data.frame and still have a data.frame that makes sense. Multiplying rows would not make sense either, since within one row you will have different types of data. Unless you have a data.frame that is all numeric, but in that case one should probably be using a matrix in the first place.