= Reconstructing a linear program from a final simplex tableau
{title2=$B=(B^{-1})^{-1}$}
For a maximization problem with unpriced unit <slack variables>, their columns in a final <simplex tableau> give $B^{-1}$, where $B$ consists of the original basic columns. The constraint right side is $\bar b=B^{-1}b$, so $b=B\bar b$. If the objective row uses <reduced costs> $r_j=c_j-c_B^TB^{-1}A_j$, its slack entries are $-y^T$ with $y^T=c_B^TB^{-1}$. Therefore $c_B^T=y^TB$. In a homogeneous linear objective with no constant offset, the displayed basic objective value must equal $c_B^T\bar b=y^Tb$. This identity detects inconsistent tableau constants; an affine offset affects the value but not the reduced costs.
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