Invertibility

Let be an matrix
suppose that . Prove that is not invertible.

I know that for to be invertible it must satisfy:
1) , such that which would shows that it's 1-1.
2) , such that , showing that it's onto.

But since even though , stipulating that the cancellation property for multiplication is not valid.

Is this right, or am I missing a large part?

Let us suppose that A is non zero and is invertible and arrive at a contradiction.

If is invertible then there exists a non zero matrix B such that
where is the identity matrix. But this would imply

a contradiction.