I think this is not getting answers because it looks like you're trying to address a well-known problem using an approach that doesn't seem very appropriate.
I see the "self-study" tag so it may be that you're just learning about the techniques... so that's the angle I'm taking when answering.
It looks like you're ignoring the fact that this is a recommender system, and just addressing the problem under the hypothesis: A's ratings can be predicted by a linear combination of the ratings by users B,C, and D.
In this case you just have a standard linear regression problem with 4 datapoints (A's existing ratings). Now I have a feeling that a regression of 4 parameters from 4 datapoints will not give you very good results... just not enough data.... and what you'll get as a result will not be useful to predict the rest of the ratings.
In a recommender system where ratings by one user are predicted from ratings by other users on the same item (movie, etc.), you're doing (a basic form of) user-user collaborative filtering, and the standard approach is that you predict user A's ratings from the top-k most similar other users.
Therefore, your weights are either uniform (1/k) (you take the mean of those ratings) (or 0 if the user is not among the top-k most similar), or dependent on the similarity (ratings from user Z are weighted according to the similarity between A and Z, calculated from other movies they've both rated).
You can take a look at this paper on collaborative filtering, it may help.