Embedding SUMPRODUCT & SUMIF

Hi there, I have a need to do a calculation based on a few criteria. I can do a simple SUMIF to get one portion of it working correct, but when I try to add in additional details to pull a specific date range of that data, I'm hitting a brick wall. (The spreadsheet I am using is a full array of columns up to CC, with over 200k rows, so I didn't want to upload such a massive file, so the attached is a simplified version)

Column A: Client Name
Column B: Total $$
Column C: Invoice Date

What I want to do is take the following =SUMIF(A2:A100000,"Client Name", B2:B100000) that gives me the total sum of Column B, and add in functionality from a date filter as well. For example, if I do a =SUMPRODUCT((C2:C100000>=DATEVALUE("1/1/2010"))*(C2:C100000<=DATEVALUE("31/1/2010")),B2:B100000) I can get the total for all clients for the date range, but I'm at a loss as to how to incorporate both of these together, so I can get the total for a client for a given month.

Any help would be greatly appreciated!

Thanks!