Page 53 - 35Linear Algebra

2.3 Elementary Row Operations 53

Example 22 (Performing EROs with Matrices)

     
0 1 0 0 1 1 7 2 0 0 4
1 0 0 2 0 0 4 = 0 1 1 7
     
0 0 1 0 0 1 4 0 0 1 4

∼
1 0 0 2 0 0 4 1 0 0 2
     
2
0 1 0 0 1 1 7 = 0 1 1 7
     
0 0 1 0 0 1 4 0 0 1 4
∼

     
1 0 0 1 0 0 2 1 0 0 2
0 1 −1 0 1 1 7 = 0 1 0 3
     
0 0 1 0 0 1 4 0 0 1 4

Here we have multiplied the augmented matrix with the matrices that acted on rows
listed on the right of example 21.

Realizing EROs as matrices allows us to give a concrete notion of “di-
viding by a matrix”; we can now perform manipulations on both sides of an
equation in a familiar way:

Example 23 (Undoing A in Ax = b slowly, for A = 6 = 3 · 2)

6x = 12

⇔ 3 −1 6x = 3 −1 12
⇔ 2x = 4
⇔ 2 −1 2x = 2 −1 4

⇔ 1x = 2

The matrices corresponding to EROs undo a matrix step by step.

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