# Detached coefficient form

Given the system $LaTeX: Ax=b$, we augment the objective equation: $LaTeX: c^{T}x - z = 0$. Then, detaching the variables, the form is:

$LaTeX: \left[ \begin{array}{rrr} A & 0 & b \\ c & {-1} & 0 \end{array} \right].$

Upon multiplying $LaTeX: Ax = b$ by the inverse of a basis, say $LaTeX: B$, where $LaTeX: A=[B \; N]$, the detached coefficient form becomes

$LaTeX: \left[ \begin{array}{ccrc} I & B^{-1}N & 0 & B^{-1}b \\ c_B & c_N & -1 & 0 \end{array} \right]$,

where $LaTeX: c=[c_{B} \; c_{N}]$ (conformal with the partition of the columns of $LaTeX: A$). Now if we subtract $LaTeX: c_B$ times the first $LaTeX: m$ rows from the last row, this becomes

$LaTeX: \left[ \begin{array}{ccrc} I & B^{-1}N & 0 & B^{-1}b \\ 0 & c_N - c_B B^{-1}N & -1 & -c_B B^{-1} b \end{array} \right]$

Recognizing that $LaTeX: B^{-1}b$ is the vector of basic levels, and the associated objective value $LaTeX: (z)$ is $LaTeX: c_B B^{-1}b$, we can drop the first $LaTeX: m$ columns plus column $LaTeX: n+1$ (corresponding to $LaTeX: z$) and form the tableau associated with this basis (adding labels to identify the basic variables in the rows and nonbasic variables in the columns).