Page 52 - 35Linear Algebra

52 Systems of Linear Equations

2.3 Elementary Row Operations

Elementary row operations are systems of linear equations relating the old
and new rows in Gaussian elimination:

Example 21 (Keeping track of EROs with equations between rows)
0
We refer to the new kth row as R and the old kth row as R k .
k
0
R =0R 1 + R 2 +0R 3
1
0
2
  R = R 1 +0R 2 +0R 3    0     
0 1 1 7 0 2 0 0 4 R 0 1 0 R 1
R =0R 1 +0R 2 + R 3 1
3
2 0 0 4 ∼ 0 1 1 7 R = 1 0 0
     0     R 2 
2
0 0 1 4 0 0 1 4 R 3 0 0 0 1 R 3
0 1
R = R 1 +0R 2 +0R 3
1 2
0
R =0R 1 + R 2 +0R 3    0   1   
2
0 1 0 0 2 R 0 0 R 1
R =0R 1 +0R 2 + R 3 1 2
3
∼  0 1 1 7   R 2 =  0 1 0   R 2 
0 
0 0 1 4 R 0 0 0 1
3 R 3
0
R = R 1 +0R 2 +0R 3
1
0
R =0R 1 + R 2 − R 3    0     
2
0 1 0 0 2 R 1 0 0 R 1
R =0R 1 +0R 2 + R 3 1
3
∼  0 1 0 3   R 0  =  0 1 −1   R 2 
2
0 0 1 4 R 0 0 0 1 R 3
3
On the right, we have listed the relations between old and new rows in matrix notation.
Reading homework: problem 3

2.3.1 EROs and Matrices

Interestingly, the matrix that describes the relationship between old and new
rows performs the corresponding ERO on the augmented matrix.

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