How to identify a demonstrative/relative pronoun e.g. in "so läßt sich die dem Element A"

Question

I am trying to self-teach scientific German for historical purposes. A sentence in a mathematical book (1922) occurs like:

Ist nunmehr eine Darstellung Γ von höherem Grade gegeben, so läßt sich die dem Element A entsprechende Matrix transformieren auf die Form:

DeepL and Google translate it the same way:

If now a representation Γ of higher degree is given, then the matrix corresponding to the element A can be transformed to the form:

My question is with the construction, so läßt sich die dem Element A. How do we understand die and dem occurring together? To which noun is die referring to? I feel die must be a demonstrative pronoun because the verb transformieren is not in the end. What could be literal translation of the so läßt sich die dem Element A entsprechende Matrix transformieren auf die Form:?

Answer 1

Both translations are ok, although the google one is closer to the original. As for "die": it is neither demonstrative (Demonstrativpronomen) nor relative (Relativpronomen), just the definite article (bestimmter Artikel).

The whole sentence is made of two main clauses (Hauptsätze) separated by a comma. That both clauses are main clauses can be seen by the fact that both sentences do have their conjugated verb not in end position. Furthermore, "transformieren" is not the conjugated verb in the second clause, it is "lässt" a modal verb that is here combined with the indicative "transformieren".

As for the phrase "die dem Element A entsprechende Matrix":

It is similar to a relative clause: "..., so läßt sich die Matrix, die dem Element A entspricht, transformieren auf die Form:" But instead of this formulation using a relative clause, the author has used present participle (Partizip Präsens).