Fundamentals of Matrix Algebra, 2011a

2R 2 + R 3 → R 3
1.3 Elementary Row Operaons and Gaussian Eliminaon
⎡
1 2 1 5
⎤
0
⎣ 0 0 1 1 2 ⎦
0 0 0 0 0
Our aenon now shis over one more column and down one row to the posion
indicated by the box; we wish to make this a 1. Of course, there is no way to do this,
so we are done with the forward steps.
Our next goal is to put a 0 above each of the leading 1s (in this case there is only
one leading 1 to deal with).
−R 2 + R 1 → R 1
This final matrix is in reduced row echelon form. .
⎡
⎣ 1 2 0 4 −2
0 0 1 1 2
0 0 0 0 0
⎤
⎦
. Example 6 Put the matrix
⎡
⎣ 1 2 1 3 ⎤
2 1 1 1 ⎦
3 3 2 1
into reduced row echelon form.
.
S
Here we will show all steps without explaining each one.
⎡
−2R 1 + R 2 → R 2
⎣ 1 2 1 3 ⎤
0 −3 −1 −5 ⎦
−3R 1 + R 3 → R 3
0 −3 −1 −8
⎡
⎤
1 2 1 3
− 1 R2 → R2
⎣
3
0 1 1/3 5/3 ⎦
0 −3 −1 −8
⎡
3R 2 + R 3 → R 3
⎣ 1 2 1 3 ⎤
0 1 1/3 5/3 ⎦
0 0 0 −3
⎡
− 1 R 3 3 → R 3
⎣ 1 2 1 3 ⎤
0 1 1/3 5/3 ⎦
0 0 0 1
⎡
−3R 3 + R 1 → R 1
⎣ 1 2 1 0 ⎤
− 5 R3 + R2 →
0 1 1/3 0 ⎦
R2
3
0 0 0 1
⎡
−2R 2 + R 1 → R 1
⎣ 1 0 1/3 0 ⎤
0 1 1/3 0 ⎦
0 0 0 1
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