Is it possible that while the simplex algorithm is working, we get a lambda $$ (Ax < b, \max c^T x )$$ $$ \lambda_B = c^T A^{-1}_B $$
with only zeros in it ? if so what does it would represent/ mean ? I think that if there is one zero, that means that the solution has one degree of freedom, so a face of degree 1 is a solution.
I'm doubtfull why in the algorithm, we allow lambda to be equal to 0 at some entries.
The number of zeros represents indeed the number of dimension of the optimal solution. I had a problem within my resolution of the problem, that's why I got a $$ \lambda \equiv 0 $$ In any other case, that shoudln't be the case, unless the whole initial set is a solution to the problem .
E.G. : an optimisation problem where the objective function is $$ \min \textbf{0} \cdot c^t $$