This should be a comment but is too long. The decidability of the question whether a finite set R of relations implies some other relation r=1 in all linear groups is the same as asking if R implies r=1 in all finite groups because of Malcev's theorem on residual finiteness of linear groups. A beautiful theorem of Slobodoskoii says this latter problem is undecidable.

A consequence is that the first order theory for finite dimensional modules over an algebra of wild representation type is undecidable. (Most books on finite dimensional algebras mistakenly assert it follows from undecidability of the word problem for groups, but one must then allow infinite dimensional modules.)

This should be a comment but is too long. The decidability of the question whether a finite set R of relations implies some other relation r=1 in all linear groups is the same as asking if R implies r=1 in all finite groups because of Malcev's theorem on residual finiteness of linear groups. A beautiful theorem of Slobodoskoii says this latter problem is undecidable.

A consequence is that the first order theory for finite dimensional modules over an algebra of wild representation type is undecidable. (Most books on finite dimensional algebras mistakenly assert it follows from undecidability of the word problem for groups, but one must then allow infinite dimensional modules.)

Edit. The question is not entirely well posed. It is impossible to decide whether your group is finitely presented as @HJRW points out. @IgorRivkin says he wants to know how to prune relations, that is, given a finite set of relations that are true remove others that are consequence. The ambiguity is whether one means redundant in a linear group or in all groups. If one means in all groups, then the fact that your group is linear seems a red herring and you are in the situation that @JoelHamkins and @DerekHolt give in their answers. My answer was assuming that you want to know if certain relations were consequences of others in all linear groups. If this is what you want the best semi-procedure is to look at images under congruence homomorphisms to rule out certain relations as being redundant and follow Derek's procedure to say that certain relations are redundant.

This should be a comment but is too long. The decidability of the question whether a finite set R of relations implies some other relation r=1 in all linear grous is the same as asking if R implies r=1 in all finite groups because of Malcev's theorem on residual finiteness of linear groups. A beautiful theorem of Slobodoskoii says this latter problem is undecidable.

A consequence is that the first order theory for modules over an algebra of wild representation type is undecidable.

This should be a comment but is too long. The decidability of the question whether a finite set R of relations implies some other relation r=1 in all linear groups is the same as asking if R implies r=1 in all finite groups because of Malcev's theorem on residual finiteness of linear groups. A beautiful theorem of Slobodoskoii says this latter problem is undecidable.

A consequence is that the first order theory for finite dimensional modules over an algebra of wild representation type is undecidable. (Most books on finite dimensional algebras mistakenly assert it follows from undecidability of the word problem for groups, but one must then allow infinite dimensional modules.)