Page 220 - Advanced engineering mathematics

200 CHAPTER 7 Matrices and Linear Systems

First, interchange rows two and three of A to form
⎛ ⎞
−2 1 6 −3
B = ⎝ 0 9 3 −7 ⎠ .
1 1 2 5
Perform this row operation on I 3 to obtain
⎛ ⎞
1 0 0
E 1 = 0 0 1 ⎠ .
⎝
0 1 0
Then
⎛ ⎞⎛ ⎞ ⎛ ⎞
100 −2 1 6 −3 −2 1 6 −3
E 1 A = 001 ⎠⎝ 1 1 2 5 ⎠ = ⎝ 0 9 3 −7 ⎠ = B.
⎝
010 0 9 3 −7 1 1 2 5
Next multiply row three of A by 7 to form
⎛ ⎞
−2 1 6 −3
C = ⎝ 1 1 2 5 ⎠ .
0 63 21 −49
Perform this row operation on I 3 to obtain
⎛ ⎞
1 0 0
E 2 = 0 1 0 ⎠ .
⎝
0 0 7
Then
⎛ ⎞⎛ ⎞ ⎛ ⎞
100 −2 1 6 −3 −2 1 6 −3
E 2 A = 010 ⎠⎝ 1 1 2 5 ⎠ = ⎝ 1 1 2 5 ⎠ = C.
⎝
007 0 9 3 −7 0 63 21 −49
Finally, add 2 times row one to row two to form
⎛ ⎞
−21 6 −3
D = −3 3 14 −1 ⎠ .
⎝
0 9 3 −7
This operation can be achieved by the elementary matrix
⎛ ⎞
1 0 0
E 3 = 2 1 0 ⎠ .
⎝
0 0 1
As a check,
⎛ ⎞⎛ ⎞ ⎛ ⎞
100 −2 1 6 −3 −2 1 6 −3
E 3 A = 210 ⎠⎝ 1 1 2 5 ⎠ = −3 3 14 −1 ⎠ = D.
⎝
⎝
001 0 9 3 −7 0 9 3 −7
This result has an important consequence. Suppose we form B from A by performing a
sequence of elementary row operations in succession. That is, we perform operation O 1 on A to
obtain A 1 , then O 2 on A 1 to form A 2 , and so on until we perform O r on A r−1 to form A r = B.We
may envision this process
O 1 O 2 O 3
A −→ A 1 −→ A 2 −→ A 3 →
O r−1 O r
··· −−→ A r−1 −→ A r = B.
We can perform each elementary operation O j by multiplying on the left by the elementary
matrix E j formed by performing that operation on I n . Then

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:23 THM/NEIL Page-200 27410_07_ch07_p187-246