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In the next example, we will first put the matrix into row-echelon form and then
write it in reduced row-echelon form.
EXAMPLE 4
Using Gauss-Jordan elimination to solve a system of equations
Solve the following system of equations using Gauss-Jordan elimination:
1 x y 2z 9
2 x y 2z 7
3 x 2y z 6
Solution
The given system of equations is first written in augmented matrix form.
1 1 2 9
£ 1 1 2 † 7 §
1 2 1 6
Step 1: Write the given augmented matrix in row-echelon form.
1 1 2
£ 0 2 4
0 3 3
1 1 2
£ 0 1 2
0 3 3
†
†
9
16 §
15
9
8 §
15
row 1 row 2
1 (row 1) row 3
1 (row 2)
2
1 1 2
£ 0 1 2
0 0 3
3 (row 2) row 3
The original matrix is now in row-echelon form.
Step 2: Write the matrix in reduced row-echelon form.
First change the leading 3 in row 3 to a 1.
1 1 2
£ 0 1 2
0 0 1
(row 3)
Use elementary row operations to obtain 0 in the third column for all entries but
the third row.
1 1 0
£ 0 1 0
0 0 1
9
† 8 §
9
9
† 8 §
3
3
† 2 §
3
1
3
2 (row 3) row 1
2 (row 3) row 2
NEL
VECTOR APPENDIX—GAUSS-JORDAN ELIMINATION 593