CJam, (削除) 22 (削除ここまで) 21 bytes
Saved 1 byte thanks to Martin Ender
{_,({0\f+}*ee::m<:.+}
Anonymous block expecting the argument on the stack and leaves the result on the stack.
How it works
_ e# Duplicate the matrix
,( e# Get its length (# of rows) minus 1
{0\f+}* e# Prepend that many 0s to each row
ee e# Enumerate; map each row to [index, row]
::m< e# Rotate each row left a number of spaces equal to its index
:.+ e# Sum each column
Business Cat
- 9.2k
- 1
- 18
- 37