Small bit of help needed?

Daniel S

How do you do Gausian elimination when you have a 3x4 matrix (taken from a simultaneous equation)? I can do 3x3's no bother, but what should I do when I get to the last line? Should I multiply the row to cancel it out? Seems a bit pointless especially since it can be solved just using the first three lines (which we were asked to do first: x=7 y=-1 z=-2)

Original question:
| 1 -5 3 | 6 |
| 3 -14 8 | 19 |
|-5 17 -14| -24|
|-1 -1 1 | -8 |

This is as far as I can go:
| 1 -5 3| 6 |
| 0 1 -1| 1 |
| 0 0 1 | -13/6|
| 0 0 ? | ? |

Daniel S

Continued eliminating below the ones:

| 1 0 0 | 20/3 |
| 0 1 0 | -7/6 |
| 0 0 1 | -13/6|
| 0 0 0 | -1/3 |

Does that mean X=20/3 Y=-7/6 Z=-13/6 and I just ignore the last line?

ride-the-spiral

What the last line says is that 0=-1/3, which either means the the system has no solutions, or that you've done the elimination incorrectly. Even though the system can be solved only using the first three you should still use the fourth as it will 1) tell you if you've made a mistake by not getting a row of zeros and 2) when you go beyond treating matrices and ways to solve linear systems of equations you'll have to know how to gaussian eliminate it anyway.

However, I checked it on wolframalpha and you should get a row of zeros at the bottom.

http://www.wolframalpha.com/input/?i=gaussian+elimination+%7B%7B1%2C-5%2C3%2C6%7D%2C%7B3%2C-14%2C8%2C19%7D%2C%7B-5%2C17%2C-14%2C-24%7D%2C%7B-1%2C-1%2C1%2C-8%7D%7D

Daniel S

ride-the-spiral wrote: »

What the last line says is that 0=-1/3, which either means the the system has no solutions, or that you've done the elimination incorrectly. Even though the system can be solved only using the first three you should still use the fourth as it will 1) tell you if you've made a mistake by not getting a row of zeros and 2) when you go beyond treating matrices and ways to solve linear systems of equations you'll have to know how to gaussian eliminate it anyway.

However, I checked it on wolframalpha and you should get a row of zeros at the bottom.

http://www.wolframalpha.com/input/?i=gaussian+elimination+%7B%7B1%2C-5%2C3%2C6%7D%2C%7B3%2C-14%2C8%2C19%7D%2C%7B-5%2C17%2C-14%2C-24%7D%2C%7B-1%2C-1%2C1%2C-8%7D%7D

The "show steps" is perfect, but is there anyway to just do the last line if you already have the first three rows done? These are the questions we have to do. Our TA said we HAVE to use Gaussian elimination.

(a) Find all solutions of each system of linear equations: 3%+3%+3%

(i)
x - 5y + 3z = 6
3x - 14y + 8z = 19
-5x + 17y - 14z = -24

(ii)
x - 5y + 3z = 6
3x - 14y + 8z = 19
-5x + 17y - 14z = -24
-x - y + z = -8

(iii)
x - 5y + 3z + 2f = 3
3x - 14y + 8z + 5f = 8

As you can see, step (ii) should be just a short one on top of (i), but I can't think of any otherway of doing it apart from doing it all out again.

Daniel S

Got most of it, but still can't find the "quick link" between part (i) and part (ii). I still need to do the whole thing out again to answer it.

(i) = [ 7, -1, -2 ]^T
(ii) = [ 7, -1, -2 ]^T
(iii) [-2 , 2t1 + 3t2 , -1 + t1 + t2, t1 , t2 ]^T

Eliot Rosewater

Daniel S wrote: »

Got most of it, but still can't find the "quick link" between part (i) and part (ii). I still need to do the whole thing out again to answer it.

At the end of part (i) you'll have something like



Could you then just stick in your new row on the bottom like so:



And just work from there?

ride-the-spiral wrote: »

However, I checked it on wolframalpha and you should get a row of zeros at the bottom.

http://www.wolframalpha.com/input/?i=gaussian+elimination+%7B%7B1%2C-5%2C3%2C6%7D%2C%7B3%2C-14%2C8%2C19%7D%2C%7B-5%2C17%2C-14%2C-24%7D%2C%7B-1%2C-1%2C1%2C-8%7D%7D

Wolfram Alpha never ceases to impress!

Daniel S

Eliot Rosewater wrote: »

At the end of part (i) you'll have something like



Could you then just stick in your new row on the bottom like so:



And just work from there?




Wolfram Alpha never ceases to impress!

Ah! So I then add the bottom line by the top line, then the second line to the bottom line, then the third line? The ones should cancel and (-8+7-1+2)=0!!! YAY!!!

Time to do all the exam papers!!!

ride-the-spiral

Yeup that works, although even if you have the first three lines done and the fourth seems superfluous you should do it, as if you don't get a row of zeros (pivot in the last column) then the system has no solution, even if it would if it were just the first three.