SUMIF across multiple columns and rows

I have a data set that shows dates across the columns and PO #s down rows. The same PO number can be used several times down the row as several items can be on the same PO.

What I would like to do is add all values from a PO across a date range. This should be in a formula, not a pivot table.

Attached Image of data set

Dataset.JPG

Attached example spreadsheet

I can solve this by stringing together several IF(SUMIF()) formulas as indicated however with a large amount of rows and columns, this is not practical. A dummy row or column to subtotal the various PO's is not practical either.

I am looking for something more eloquent.

In my example I want to add all PO values for PO 10 from Jan 2 to Jan 4th.

My actual spreadsheet will have multiple POs being added over multiple date ranges (date ranges can be different for each PO)