Reconstructing a linear program from a final simplex tableau ($B=(B^{-1}{)}^{-1}$)

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 $r_j=c_j-c_B^T{B}^{-1}{A}_j$, its slack entries are $-y^T$ with $y^T=c_B^T{B}^{-1}$. Therefore $c_B^T=y^{T}B$. In a homogeneous linear objective with no constant offset, the displayed basic objective value must equal $c_B^T\bar{b}=y^{T}b$. This identity detects inconsistent tableau constants; an affine offset affects the value but not the reduced costs.