Page 17 - Matrix Analysis & Applied Linear Algebra

1.2 Gaussian Elimination and Matrices 9

• Type II: Replace row i by a nonzero multiple of itself to produce
 
M 1∗
.
 . . 
 
 αM i∗  , where α =0. (1.2.8)


.
 
 . . 
M m∗
• Type III: Replace row j by a combination of itself plus a multiple of row
i to produce
 
M 1∗
.
 . . 
 
 
M i∗
 
 . 
.
 .  . (1.2.9)
 
 
 M j∗ + αM i∗ 
.
 
.
 . 
M m∗
To solve the system (1.2.4) by using elementary row operations, start with
the associated augmented matrix [A|b] and triangularize the coefficient matrix
A by performing exactly the same sequence of row operations that corresponds
to the elementary operations executed on the equations themselves:
   
11 1 2 1 1 1
2
-1
 621 −1  R 2 − 3R 1 −→  0 −2 −4 
−221 7 R 3 + R 1 0 3 2 8 R 3 +3R 2
 
2 1 1 1
−→  0 −1 −2 −4  .
0 0 −4 −4
The final array represents the triangular system
2x + y + z = 1,
− y − 2z = − 4,
− 4z = − 4
that is solved by back substitution as described earlier. In general, if an n × n
system has been triangularized to the form
 
t 11 t 12 ··· t 1n c 1
 0 t 22 ··· t 2n c 2 
 . . . . .  (1.2.10)
 . . . . . 
. . . . .
0 0 ··· t nn c n
in which each t ii = 0 (i.e., there are no zero pivots), then the general algorithm
for back substitution is as follows.