Let $B\_{n\times n}$ be a nilpotent matrix. How should I go around proving that $B$ is not invertible?

I was thinking this: if $B\*B = 0$ , then if I rank $B$ to echelon form, I will always have two or more rows that are the same, and then because $B$ is square, it is not possible to find another matrix $(B^{-1})$ that will transform $B$ to $I\_{n}$...   
But I don't know how to write it formally.