What do you mean by , because we don't know what "y" is! I think maybe you're referring to the case where x=y. And I think in this case will have the full column rank if , and it will have the full row rank if .
Since is square, and it is an nxn matrix, it follows that is invertible if and only if has rank n. However has the same rank as x itself, so is invertible if rank(x)=n (this means if x has full column rank).
Again I'm assuming x=y.