Without multiplying matrices, find bases for the row and column spaces of A:
How do you know from these shapes that A cannot be invertible?
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?