Let $B\in \mathbb{C}^{n\times m}$, where $n\geq m$, prove that $B^{*}B$ is invertible if and only if $B$ is invertible.

Question

My proof: forward direction: $$(B^{*}B)(B^{*}B)^{-1}=I$$ $$B^{*}B(B^{-1})^{*}B^{-1}=I$$ Only possible when $BB^{-1}=I$. Now, for the backward direction: the same way.