1. ## Invertibility

Let $A$ be an $n \times {n}$ matrix
suppose that $A^2 = 0$. Prove that $A$ is not invertible.

I know that for $A$ to be invertible it must satisfy:
1) $T:A \rightarrow B$, such that $UT = I_{v}$ which would shows that it's 1-1.
2) $U:B \rightarrow A$, such that $TU = I_{w}$, showing that it's onto.

But since $A^2 =0$ even though $A \neq 0$, stipulating that the cancellation property for multiplication is not valid.

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

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

If $A$ is invertible then there exists a non zero matrix B such that
$AB = I_n = BA$ where $I$ is the identity matrix. But this would imply
$A = AI = A(AB) = A^2 B = 0B = 0,$