I want to generate a random matrix $V\in [0,1]^{(n+1)\times n}$(not necessarily being binary ), such that for each row of $V$, there is at least one component is $1,$ and at least one component is $0$. Moreover. the constructed $A:=[V,1^{n+1}]$(concatenate V with a column valued 1), is invertible.
Any suggestions ?
Thank you in advance