Help with IF and Sum functions

I have a spreadsheet with a list of names spread throughout a column (Column A) and some numbers in the next few column (Column B, C, D, E) I'm trying to set up in Column F the ability to use the IF function to pick out certain names (Mike F for example) then add up the corresponding numbers found in the columns next to that name. Now Mike may be listed several times in Col A the way I've set up the statement it's adding up all the numbers in my given range no matter what name is in Col A. Here is the way I've got the function set up:

=if(A4:A200="Mike F", Sum(B4:B200)+sum(C4:C200)+sum(D4:D200)+sum(E4:E200),0)

Is there another way to accomplish what I'm trying to do or can it not be done because it doesn't seem to be looking for Mike in Col A, if I put a number next to Andrew it returns that number instead of 0.
Any help with this is greatly appreciated.
Thanks
Bob

Don't know if my original response got thru so here it goes:
I tried your suggestion but the cell ended up saying #N/A "Error Array arguments to MULTIPLY are of different size."
When I changed the =SUMPRODUCT((A4:A7=A10)*B4:E7) the B4:E7 code to B4:F4 it seemed to work.
I've included the spreadsheet to see how it's set up

You have to have equal number of rows in both ranges of the formula.

You have
=SUMPRODUCT((A4:A7=J16)*B4:E12)

One range is rows 4:7, the other is 4:12
They need to both be the same

So you probably want
=SUMPRODUCT((A4:A12=J16)*B4:E12)

Thank you so much!! It works perfectly for what I need it show me.
Thanks for your help.