This is a true and false question. The actual question reads:
"In solving a linear program by the simplex method, starting with a feasible tableau, a different feasible point is generated after every pivot step"
This is what I have written down as the answer so far;
The statement is true. Starting from a feasible tableau, after every pivot step we would get a better minimized/maximized (depending on what the question asks for) value of the objective function. Thus, we would get a different feasible point after every pivot until we can not pivot anymore; in which at that point the tableau is optimal.
Is my answer correct? Any help would be much appreciated!
Your answer is only correct if the feasible region is simplicial, meaning that if you are working in dimension $d$ then every feasible tableau can be obtained once and only once as an intersection of $d$ equations. Otherwise, it is possible that when you pivot, you will get the same answer you had before, but now as an intersection of different equations, and in fact it is even possible (if you're not careful with your pivot rule) that you will get a cycle and thus not even converge to an optimal solution. This was the case for the original pivot rule proposed, and it took a bit of time to find a pivot rule that provably had no cycles.