Usually, I do my matrix multiplication like this:
[1 0 0 dx] | [px] [px + dx]
[0 1 0 dy] | [py] [py + dy]
[0 0 1 dz] | [pz] = [pz + dz]
[A B C 1] | [1] [1]
Where the translation is along the right-hand edge of the matrix.
Once, I forgot about that and put them on the lower line instead, on the places marked as A B C
above.
What did that do to the resulting points? With the upper-left 3x3 being an identity matrix, and the translation being zero, nothing happened to my point
[1 0 0 0] | [1] [1]
[0 1 0 0] | [1] [1]
[0 0 1 0] | [1] = [1]
[1 2 3 1] | [1] [7]
except that 1 we always use in fourth place gets changed. Does this mean further multiplication with the point results in a mess? What am I doing by changing the bottom row?