Upon inspection, I find that a basis for the row space of A is (3,0,3) and (1,1,2) while a basis for the column space is (1,4,2) and (2,5,7). Although I'm not sure why...
If we write A= BC, then C maps all of into a subspace of which can, of course, have dimension no larger than 2. B then maps that subspace into a subspace of R^3 which can have dimension no larger than 2. Since A= BC maps into a two dimensional subspace of , it cannot be invertible.
How do you know from these shapes that A cannot be invertible?