Using an Augmented Matrix

Date: 11/21/1999 at 16:31:02
From: Katie
Subject: Augmented matrices

I'm really stuck on how to solve augmented matrices. I understand how 
you solve for 1 in cell 11 and I understand you have to add row 1 and 
row 2 to get a new row 2, but I don't understand what to multiply when 
you are adding the two equations. The problem is this:

      x + 3y +  z = 3
      x + 5y + 5z = 1 
     2x + 6y + 3z = 8

so the matrix becomes

     [1 3 1 | 3]
     [1 5 5 | 1]
     [2 6 3 | 8]

So then you add row 1 [1 3 1 | 3] to row 2 [1 5 5 | 1], but what does 
row 1 get multiplied by?

Date: 11/21/1999 at 18:30:17
From: Doctor Anthony
Subject: Re: Augmented matrices

     [ 1   3   1   |   3 ]     r1
     [ 1   5   5   |   1 ]     r2
     [ 2   6   3   |   8 ]     r3

r2 = r2-r1

     [ 1   3   1   |   3 ]     r1
     [ 0   2   4   |  -2 ]     r2
     [ 2   6   3   |   8 ]     r3

r3 = r3-2r1

     [ 1   3   1   |   3 ]     r1
     [ 0   2   4   |  -2 ]     r2
     [ 0   0   1   |   2 ]     r3

r2 = r2/2

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

At this stage we can see

                           z = 2

         y + 4 = -1   so   y = -5

    x - 15 + 2 = 3    so   x = 16

The solution set is x = 16, y = -5, z = 2

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