Count the number of operations needed to perform Gauss-Jordan elimination—that is, to re...

Count the number of operations needed to perform Gauss-Jordan elimination—that is, to reduce [A | b] to its reduced row echelon form

(where the zeros are introduced into each column immediately after the leading 1 is created in that column). What do your answers suggest about the relative efficiency of the two algorithms?

We will now attempt to analyze the algorithms in a general, systematic way. Suppose the augmented matrix [A | b] arises from a linear system with n equations and n variables; thus,

We will assume that row interchanges are never needed—that we can always create a leading 1 from a pivot by dividing by the pivot.