Multiplying the contents of a cell by a multiplier, leaving blank cells blank.

I'll try not to be long winded here but I've been working on this for hours and am stumped:

I have a bunch of data (128x128 grid) on Sheet1.
On Sheet 2 I need the data from sheet1 to be multiplied by a multiplier (unique multiplier per each row*)
*Sheet2 Column EA contains the multipliers

I have tons of blank cells on Sheet1 and need to have the blanks carry over to Sheet2.
Another caveat is that several of the multipliers in Column EA are 0.000.
When that is the case I need the data in that row to show 0.000 only if data is filled in a cell--again all other cells would remain blank.

I have experimented with Paste special (by copying all data from sheet1 to sheet2) and have tried to write a formula that makes this work but I haven't figured it out yet.

ANY advice or ideas? :/

Thank you! I hope this helps.
I just took a very small sample of my workbook.
Sheet 1 has the current data.
Sheet 2 is where I want to multiply the exact numbers from Sheet 1 by the light green column (G in this example)
Sheet 3 is what I would like it to look like when it is done. (I manually coded each cell by placing an = in front of the current cell content and then added *G(row#)

I would like to have a way to obtain the results on Sheet 3 without having to enter each cell manually. Thank you!!

sampledataOWP.xlsx

Thank you!
I am still getting zeros straight across my rows...is there another setting I need to change?

Okay I just noticed that attachment!
I am trying to figure out if there is a way to keep cells that were originally blank, blank.
I believe there I would have to create an If statement but I'm not sure how.
Basically: IF the cells are blank...do nothing.
IF they are not...I need to do the equation.

Thank you so much!
I have learned so much about excel in one week my head is going to explode!