Construct a linear programming problem for which both the primal and the dual problem has no feasible solution
Let $A=\left(\begin{smallmatrix} -1&0\\0&1\end{smallmatrix}\right)$, $b=\left(\begin{smallmatrix}1\\1\end{smallmatrix}\right)=-c$. $Ax\ge b$ and $A^Ty\le c$ cannot both be satisfied with positive $x,y$.