Hello everyone,

I am having trouble understanding one line of the proof for a claim that involves degeneracy in the simplex method, all of which are posted below. Idicated by the red arrow is the step that troubles me: "also $\displaystyle a_{ij}^{-1} = 0 $".

I have also written my thoughts below.

Thank you very much!

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Thoughts:

I know the following from the given fact that the objective value doesn't change after one application of the simplex method:

$\displaystyle \displaystyle q = \frac{-p_j}{a_{ij}} \times b_i + q \Rightarrow$

$\displaystyle \displaystyle \frac{-p_j}{a_{ij}} \times b_i = 0 \Rightarrow \frac{b_i}{a_{ij}} = 0$, since $\displaystyle \displaystyle p_j \neq 0$. However, how does the step in red follow from this?

Actual Question and Proof: