
The procedure for GuassJordan Elimination is as follows:
 Find the leftmost column that is not all zeros and swap its row with the top row.
 Make the leading entry in the first row a "1" (If the top entry is a, then multiply the top row by 1/a).
 Use the top row to make all the entries in the column below the leading one be zero.
 Ignoring the top row, repeat steps 1 to 4 until there are no more leading ones.
 Finally, use each leading 1 to make all entries on the column above the zeros.
Example:
Solve for x, y, and z in:Solution:
2y  3z = 2 2x + z = 3 x  y + 3z = 1 This is now in reduced rowechelon form, so we can read off the answer:
é
ê
ë0
2
12
0
13
1
3ï
ï
ï2
3
1ù
ú
ûwrite the system as an augmented matrix
é
ê
ë1
2
01
0
23
1
3ï
ï
ï1
3
2ù
ú
ûinterchange first and third row
(to make top left entry nonzero)
é
ê
ë1
0
01
2
23
5
3ï
ï
ï1
1
2ù
ú
ûAdd 2 times first row to second row
(to get 0 in first column of row 2)
é
ê
ë1
0
01
1
23
5/2
3ï
ï
ï1
1/2
2ù
ú
ûDivide second row by 2
(to get a leading 1 in row 2)
é
ê
ë1
0
01
1
03
5/2
2ï
ï
ï1
1/2
1ù
ú
û
Add 2 times second row to third
(to get 0's in the second column)
é
ê
ë1
0
01
1
03
5/2
1ï
ï
ï1
1/2
1/2ù
ú
ûDivide third row by 2
(to get a leading 1)
é
ê
ë1
0
01
1
00
0
1ï
ï
ï1/2
7/4
1/2ù
ú
ûAdd 3 times third row to first row
Add 5/2 times third row to second row
(to get 0's in third column)
é
ê
ë1
0
00
1
00
0
1ï
ï
ï5/4
7/4
1/2ù
ú
ûAdd second row to first row x = 5/4 ,y = 7/4 , andz = 1/2 .Check that the answer satisfies the initial equations (in case we made arithmatic errors):
All of these check out, so our solution is correct.2y  3z = 2(7/4)  3(1/2) = 7/2  3/2 = 4/2 = 2
2x + z = 2(5/4) + 1/2 = 5/2 + 1/2 = 6/2 = 3
x  y + 3z = (5/4)  (7/4) + (1/2) = (2/4) + 3/2 = (1/2) + 3/2 = 2/2 = 1

