Suppose that we have a basic feasible solution of the system Ax = b, x ≥ 0 with basis B. Suppose that z _k – c _k > 0 and x _k is introduced into the basis and x _Br is removed from the basis. Denote the new basis by B’. Show algebraically that after pivoting:

a. The column under x is (B’) ^-1 a _j .

b. The column under the right hand side is (B’) ^-1 b.

c. The new cost row is composed of (c _B ) (B’) ^-1 a _j - c _i