INTRODUCTION TO LINEAR ALGEBRA
MATH 361
Southwestern   
Adventist University 
 
   Distance Education Lawrence E. Turner, Jr., Ph.D.  


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Gaussian Elimination − alternative


Let us apply the Gaussian Elimination Rules without trying to avoid fractions.

The original augmented matrix is:

Augmented Matrix

To make the first element a 1, we will multiply Row 1 by 1/2 (apply Rule #2)   (1/2)·R1 ⇔ R1:

Augmented Matrix

Now that we have a 1 in the first row of the first column, we will use this row and use Rule #3 to change the other values in the first column to 0's. If we multiple row 1 by −1 and add it to row 2, then the left-most value of 1 in row 2 will become a 0.

Multiply Row 1 by −1 and add it to Row 2 (apply Rule #3)  −1·R1 + R2 → R2:

−1·R1:−13/220 
R2:1214 
 
 
 07/234→ R2
Augmented Matrix

Multiply Row 1 by −3 and add it to Row 3 (apply Rule #3)  −3·R1 + R3 → R3:

−3·R1:−39/260 
R3:3132 
 
 
 011/292→ R3
Augmented Matrix

Column 1 is finished. We turn our attention to the second column and first focus on the second element which we want to become a 1.

Multiply Row 2 by 2/7 (apply Rule #2)  (2/7)·R2 → R2:

Augmented Matrix

Now we want 0's in the rest of the column.

Multiply Row 2 by 3/2 and add it to Row 1 (apply Rule #3)  (3/2)·R2 + R1 → R1:

(3/2)·R2:03/29/712/7 
R1:1−3/2−20 
 
 
 10−5/712/7→ R1

Augmented Matrix

Multiply Row 2 by −11/2 and add it to Row 3 (apply Rule #3)  (−11/2)·R2 + R3 → R3:

(−11/2)·R2:0−11/2−33/7−44/7 
R3:011/292 
 
 
 0030/7−30/7→ R3

Augmented Matrix

Now we have the first two columns finished and turn our attention to the third column and the third element in order to turn it into a 1.

Multiply Row 3 by 30/7 (apply Rule #2)  (30/7)·R3 → R3:

Augmented Matrix

Multiply Row 3 by 5/7 and add it to Row 1 (apply Rule #3)  (5/7)·R3 + R1 → R1:

(5/7)·R3:005/7−5/7 
R1:00−5/712/7 
 
 
 0001→ R1

Augmented Matrix

Multiply Row 3 by −6/7 and add it to Row 2 (apply Rule #3)  (−6/7)·R3 + R2 → R1:

(−6/7)·R3:00−6/76/7 
R2:016/78/7 
 
 
 0102→ R2

Augmented Matrix

And, we have a solution—the same solution that we found before!

Please note, for this process, we have not used Rule #1, exchange two rows. While there are times it is necessary to use Rule #1, judicious use of this rule at other times can help with the "arithmetic."

 
© 1999, 2000, 2001, 2002, 2003 by Lawrence Turner