Given an $n$ by $n$ matrix A which is singular, $x \in \mathbb{R}^n$ such that $x_i <0 \quad \forall 1 \le i \le n$. I'm wondering under which assumptionw on $A$, the following matrix $$ B=\left[\begin{array}{cc} A & x\\ 1_{1 \times n} & 0 \end{array}\right] $$ is non-singular. I'm stuck and find it rather tough when both of the matrices on the the diagonal of $B$ is singular, even though the structure of $B$ looks not too complicated.
Could anyone kindly suggest any ideas? I'd appreciate that a lot.