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1.3 Gauss–Jordan Method 15
1.3 GAUSS–JORDAN METHOD

The purpose of this section is to introduce a variation of Gaussian elimination
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that is known as the Gauss–Jordan method. The two features that dis-
tinguish the Gauss–Jordan method from standard Gaussian elimination are as
follows.

• At each step, the pivot element is forced to be 1.

• At each step, all terms above the pivot as well as all terms below the pivot
are eliminated.
In other words, if
 
a 11 a 12 ··· a 1n b 1
 a 21 a 22 ··· a 2n b 2 
 . . . . 
 . . . . .
. . . . . . 
a n1 a n2 ··· a nn b n
is the augmented matrix associated with a linear system, then elementary row
operations are used to reduce this matrix to

10 ··· 0 s 1
 
 01 ··· 0 s 2 
 . . . . .  .
 . . . .
. . . . . . 
00 ··· 1 s n
The solution then appears in the last column (i.e., x i = s i ) so that this procedure
circumvents the need to perform back substitution.
Example 1.3.1
Problem: Apply the Gauss–Jordan method to solve the following system:
2x 1 +2x 2 +6x 3 = 4,
2x 1 + x 2 +7x 3 = 6,
−2x 1 − 6x 2 − 7x 3 = − 1.
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Although there has been some confusion as to which Jordan should receive credit for this
algorithm,it now seems clear that the method was in fact introduced by a geodesist named
Wilhelm Jordan (1842–1899) and not by the more well known mathematician Marie Ennemond
Camille Jordan (1838–1922),whose name is often mistakenly associated with the technique,but
who is otherwise correctly credited with other important topics in matrix analysis,the “Jordan
canonical form” being the most notable. Wilhelm Jordan was born in southern Germany,
educated in Stuttgart,and was a professor of geodesy at the technical college in Karlsruhe.
He was a prolific writer,and he introduced his elimination scheme in the 1888 publication
Handbuch der Vermessungskunde. Interestingly,a method similar to W. Jordan’s variation
of Gaussian elimination seems to have been discovered and described independently by an
obscure Frenchman named Clasen,who appears to have published only one scientific article,
which appeared in 1888—the same year as W. Jordan’s Handbuch appeared.