You know, when , we have that . So in this case, it doesn't make sense to divide by .
With the matrix is
Which corresponds to the equations x+ y+ z= 1, 0= 0, 0= 0. It should be obvious that you can choose, say, y and z to be any numbers you want and solve for the corresponding x. That is why there are an infinite number of solutions.
As HappyJoe said (may he long be happy), you can only divide by as long as so that it is not valid to do that and then let .