Row Operations and an Augmented Matrix

Date: 07/26/99 at 15:52:41
From: Shelley Tomberlin
Subject: Linear systems of equations

Please help! I was given this class assignment and the more I look at 
it, the more confused I become. These were the instructions I was 
given:

Solve the following linear system of equations by using row operations 
on the augmented matrix of the system. Show ALL checks and write the 
final answer in ordered notation. Our professor prefers that ALL other 
numbers be zeroed out except the numbers in the diagonal position.

  a -  b      - d + e = -10
 2a -  b - 3c + d - e =   0
      2b      - d + e =  -1
-3a      +  c     + e =   5
  a -  b - 4c + d - e =   4

Thanks.

Date: 07/27/99 at 05:46:18
From: Doctor Anthony
Subject: Re: Linear systems of equations

The augmented matrix is

     [ 1   -1    0   -1    1 | -10]  r1
     | 2   -1   -3    1   -1 |   0|  r2
     | 0    2    0   -1    1 |  -1|  r3
     |-3    0    1    0    1 |   5|  r4
     [ 1   -1   -4    1   -1 |   4]  r5

   r2 = r2 + r4 + r5

     [ 1   -1    0   -1    1 | -10]  r1
     | 0   -2   -6    2   -1 |   9|  r2
     | 0    2    0   -1    1 |  -1]  r3
     |-3    0    1    0    1 |   5|  r4
     [ 1   -1   -4    1   -1 |   4]  r5

   r4 = r4 + 3.r5

     [1   -1     0   -1    1 | -10]  r1
     |0   -2    -6    2   -1 |   9|  r2
     |0    2     0   -1    1 |  -1]  r3
     |0   -3   -11    3   -2 |  17|  r4
     [1   -1    -4    1   -1 |   4]  r5

   r5 = r5 - r1

     [1   -1     0   -1    1 | -10]  r1
     |0   -2    -6    2   -1 |   9|  r2
     |0    2     0   -1    1 |  -1]  r3
     |0   -3   -11    3   -2 |  17|  r4
     [0    0    -4    2   -2 |  14]  r5

and continuing with these row operations we end up with

     [1   0   0   0   0 | -3]  r1
     |0   1   0   0   0 |  2|  r2
     |0   0   1   0   0 | -1]  r3
     |0   0   0   1   0 |  2|  r4
     [0   0   0   0   1 | -3]  r5

and so  a = -3,  b = 2,  c = -1,  d = 2,  e = -3


- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/