I already started a thread about tensors invariance but this question which seems the same is actually about something else.
I have read quite a few books where the author (I cannot find what books are) claims the reason for general relativity using tensors is that they are invariant. However, I feel this statement is just wrong. Vectors and matrices are also invariant and yet GR is not formulated in a vectorial fashion.
I guess the reason tensors and not vectors are employed as that they are completely necessary to describe some properties such as the metric tensors.
But anyway, were we not to use tensors (or particular cases such as matrices, vectors o scalars), what other mathematical objects could you possibly use? Is there anything which falls out of the category of invariant objects that can be employed to formulate a theory?