Appendix B 



Derivation of Pivoting Rules and 

 Pivot Row and Column Selection Rules 



I. Pivoting Rules 



The pivoting process is a method of solving simultaneous equations. 

 By algebraically solving two such equations and following the steps with 

 the coefficient matrix, rules for the pivoting process are developed. 



Given: two general equations, and the corresponding coefficient 

 matrix : 



Si = aXi + bXa + ki 

 S2 = cXi + dX2 + k2 



Si =: 

 S2 = 



The first step is to solve for Xj and S2 (in terms of X2, Si, a, b, c, d, 

 ki and k2). The column headed "Xj" is the pivot column. The row 

 headed "Si" is the pivot row. The element at the intersection is the 

 pivot element. 



Solving for Xi : 



aXi ^ Si — bX2 — ki 



Xi = l/a(Si) — b/a(X2) — ki/a (Equation 1) 



Solving for S2 : 



S2 = c[l/a(Si) - b/a(X2) - ki/a] + dX2 + kg 



S2 - c/a(Si) — c(b/a) (X2) — c(ki/a) + dXs + k2 



Combining terms — 



S2 = c/a(Si) + [d — c(b/a)] X2 + [k2 — c(ki/a) ] (Equation 2) 



Equations (1) and (2) yield the matrix: 



Si X2 1 



Xi = ■" 



S2 = 



From this, four rules can be drawn for the pivoting process: 



