Let $\displaystyle A$ be an $\displaystyle n \times {n}$ matrix

suppose that $\displaystyle A^2 = 0$. Prove that $\displaystyle A$ is not invertible.

I know that for $\displaystyle A$ to be invertible it must satisfy:

1) $\displaystyle T:A \rightarrow B$, such that $\displaystyle UT = I_{v}$ which would shows that it's 1-1.

2) $\displaystyle U:B \rightarrow A$, such that $\displaystyle TU = I_{w}$, showing that it's onto.

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

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