to see why this is, let's look at a matrix that effects this change. suppose you add row 1 to row 2. this matrix that does this (via left-multiplication) is:
this is a "shear matrix", it sends squares in the plane to rhombuses (which have the same area as the squares).
if we replace the 1 in the 2,1-position with an arbitrary scaling factor r, we just change the "slant" of the resulting rhombus, which doesn't affect the area (as it still has the same height perpendicular to its base).
surely you can see that det(PA) = det(P)det(A) = (1)(det(A)) = det(A).