How about
If you multiply a general vector (on the right side!) v=(x,y,z)' with the given matrix,
you will end up with the vector (x+y+z,x+y+z,x+y+z)' <-- (transposed).
Now if A reprisents a linear map h:V->W. Then ker(A) corresponds to the
subspace which image under h is 0.
Now what that means is what does v=(x,y,z) have to be so that when you multiply it
with A you get 0?