The correct answer is C. This matter should be looked into by you.
The other options are incorrect because they do not express the same level of urgency or importance as the original sentence. Option A states that the matter has already been looked into, which is not the case. Option B states that the matter may be looked into, which does not express the same level of certainty as the original sentence. Option D is grammatically incorrect.
The original sentence expresses a strong sense of obligation, which is why the correct answer is C. This matter should be looked into by you. This means that you are the only one who can or should look into this matter. It is important that you do so as soon as possible.