# simplex $B^{-1} \cdot A_j$ tableau interpretation

In an iteration of the simplex tableau implementation, what is the interpretation of the columns $$B^{-1} \cdot A_j$$ underneath each variable $$x_j$$?

Dantzig & Thapa (Section 3.5) call it the "representation of the sth activity in terms of the basic set of activities" where in your case activity $$s$$ is $$x_j$$. In other words the vector describes the change of the values of the basic variables per unit of change of the non-basic variable $$x_j$$.
Example: solving \begin{align} \text{max.}\qquad & z=5x_1+6x_2,\\ \text{s.t.}\qquad & 2x_1+3x_2 \leq 18, & \text{(1)}\\ & 2x_1+x_2 \leq 12, & \text{(2)}\\ & x_1,x_2\geq 0, \end{align} you get the tableau $$\begin{array}{cccccc} \hline x_1 & x_2 & s_1 & s_2 & z & \text{RHS}\\ \hline 0 & 1 & 0.5 & -0.5 & 0 & 3\\ 1 & 0 & -0.25 & 0.75 & 0 & 4.5\\ \hline 0 & 0 & 1.75 & 0.75 & 1 & 40.5 \\ \hline \end{array}$$ with optimal solution $$x_1=4.5$$, $$x_2=3$$. Graphically, you can see the vectors of the non-basic variables $$s_1$$ and $$s_2$$ pointing into the direction of change: 