MAX Formula Only Working for the First Cell

I am a first-time poster and am having an issue getting the MAX function to compute the correct value. The formula works for the first cell, but I want to generate a series of maximum values and it is not working for the subsequent cells. I attached a sample spreadsheet. The problem area is shaded in blue in the spreadsheet.

Here is what I am trying to do: I have values in column A, with values appearing one or more times down the column. There are numerical values in column B. I want to know the highest value in column B for each unique value in Column A. The only way I know of doing this is to dedupe Column A and then send each of the unique values through a MAX function to find the maximum value that appears for each of the column A values. This is the formula I am using:

=MAX(IF($A:$A=E2,$B:$B))

It works for E2, but when duplicating the function in the next row and looking for the max value for the team name in E3, a "0" is returned, whereas a "1" should be returned. This is what the formula in that cell looks like:

=MAX(IF($A:$A=E3,$B:$B))

What am I doing wrong? Thanks in advance.

Thank you so much! I really appreciate that you provided not only the CTRL + SHIFT + ENTER tip, but also the named range. Very helpful!

Thanks for that tip too! Just closed it out.