Gauss Jordan Matrix

12.09.2008, 18:55

Gauss Jordan Matrix

I just started taking an online math course at m university and I need help with a matrix. It's probably really easy, but I just can't figure it out! I need to find out all solutions for K.

2 , -2 , 3 = 0
-2 ,-1 ,6 = 0
1 , 2 , K = 0

Thanks,
LJ

12.09.2008, 19:03

tigerbine

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RE: Gauss Jordan Matrix
So you don't speak german? Im sorry, but I don't understand your notation. Could you write it in "matrix vector" style? Or explain, what for example

2,-2,3 =0

stands for.

Willkommen

12.09.2008, 19:38

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RE: Gauss Jordan Matrix
$\begin{eqnarray*} \begin{vmatrix} 2 & -2 & 3 = 0\\ -2 & -1 & 6 = 0 \\ 1 & 2 & alpha =0 \end{vmatrix} \end{eqnarray*}$

Doch, Ich spreche ein bisschen Deutsch, aber Ich verstehe Mathe Wörte auf Deutsch nicht so gut.

Laura

12.09.2008, 19:40

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RE: Gauss Jordan Matrix
Sorry, the "alpha" and "K" are suposed to be the same thing. Just pressed the wrong button.

Laura

12.09.2008, 19:44

tigerbine

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RE: Gauss Jordan Matrix
So, we almost got it. Do you want to solve a system of linear equations? Or do you want to calculate the matrix inverse? I don't know the name "Gauss-Jordan-Matrix". just "Gauss-elimination" or "Gauss-jordan-elimination"

http://en.wikipedia.org/wiki/Gauss%E2%80...dan_elimination
http://en.wikipedia.org/wiki/Gaussian_elimination

12.09.2008, 19:46

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RE: Gauss Jordan Matrix
Linear equation. Have to find all the values for "alpha". I am new at this whole matheboard thing. I'll get it sometime. Thanks

12.09.2008, 19:54

tigerbine

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RE: Gauss Jordan Matrix
ok, let's write it different. You want to solve $\begin{eqnarray*} Ax=0 \end{eqnarray*}$ ,

$\begin{eqnarray*} \begin{pmatrix} 2&-2&3 \\ -2 &-1 & 6 \\ 1 & 2 & K \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\x_3 \end{pmatrix} =\begin{pmatrix}0 \\ 0 \\ 0 \end{pmatrix} \end{eqnarray*}$

We know, that there is always (no matter what K) the solution:

$\begin{eqnarray*} \begin{pmatrix} x_1 \\ x_2 \\x_3 \end{pmatrix} = \begin{pmatrix}0 \\ 0 \\ 0 \end{pmatrix} \end{eqnarray*}$

Now we could ask the question, for which K there is no other solution, or for which K there are other solutions. Is that the question? (Sorry, my english is not very good.)

12.09.2008, 19:57

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RE: Gauss Jordan Matrix
Yes, that is my question. *g*

12.09.2008, 20:00

tigerbine

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RE: Gauss Jordan Matrix
So, do you know the terms "rank of a matrix", "regular matrix", "singular matrix"

12.09.2008, 20:04

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RE: Gauss Jordan Matrix
Not really. Just started the course this week and I haven't taken any math courses for 6 years so I've forgotten everything. K could be anthing couldn't it? I mean it doesn't matter since x3 is 0. I think I am just too stressed out to think straight.

12.09.2008, 20:18

tigerbine

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RE: Gauss Jordan Matrix
I'll do it for you now, but you promise to read the article "gaussian elimination", ok?

We start with

$\begin{eqnarray*} \begin{pmatrix} 2&-2&3 \\ -2 &-1 & 6 \\ 1 & 2 & K \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\x_3 \end{pmatrix} =\begin{pmatrix}0 \\ 0 \\ 0 \end{pmatrix} \end{eqnarray*}$

Now we eliminate the first column. row II' = I+II, row III' = III-0.5I

$\begin{eqnarray*} \begin{pmatrix} 2&-2&3 \\ 0 &-3 & 9 \\ 0 & 3 & K-1.5 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\x_3 \end{pmatrix} =\begin{pmatrix}0 \\ 0 \\ 0 \end{pmatrix} \end{eqnarray*}$

Now the second column. III'' = II'+III'

$\begin{eqnarray*} \begin{pmatrix} 2&-2&3 \\ 0 &-3 & 9 \\ 0 & 0 & K+7.5 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\x_3 \end{pmatrix} =\begin{pmatrix}0 \\ 0 \\ 0 \end{pmatrix} \end{eqnarray*}$

So, the matrix is regular, if its determinat (http://en.wikipedia.org/wiki/Determinant) is unequal 0. After the gauss-alforithm our matrix is an upper triangular matrix (http://en.wikipedia.org/wiki/Triangular_matrix). Its determinat is the product of the element on the diagonal

$\begin{eqnarray*} \det(A) = 2\cdot (-3) \cdot (K+7.5) \end{eqnarray*}$

If $\begin{eqnarray*} K\neq -7.5 \end{eqnarray*}$ A is regular and there is no other solution. Otherwise we have got endless solutions.

12.09.2008, 20:23

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RE: Gauss Jordan Matrix
Thank you so much! I am definately going to read that. How to do this kind of question wasn't in our text book so that is good to have somewhere that I can read to understand it.

Laura

12.09.2008, 20:28

WebFritzi

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RE: Gauss Jordan Matrix

Zitat:

Original von LJ
Yes, that is my question. *g*

So, why don't you ask that right away? unglücklich

@tigerbine: If she doesn't know the determinant, why do you come up with it?

@LJ: tigerbine has already done some work for you. She has performed the Gaussian Algorithm up to

$\begin{eqnarray*} \begin{pmatrix} 2&-2&3 \\ 0 &-3 & 9 \\ 0 & 0 & K+7.5 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\x_3 \end{pmatrix} =\begin{pmatrix}0 \\ 0 \\ 0 \end{pmatrix}. \end{eqnarray*}$

Now, you can solve the equation system as usual. We have (last equation)

$\begin{eqnarray*} (K + 7.5)x_3 = 0. \end{eqnarray*}$

What does that mean for K?

12.09.2008, 20:32

tigerbine

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RE: Gauss Jordan Matrix
@WebFritzi:

guess LJ is female. Because she wrote "forgotten" I tried do remind her of some terms. But you're right.

By the way, do you know an online dictionary "Deutsch - Englisch" for math vocabulary. I'm running out of "words" Big Laugh

12.09.2008, 20:45

WebFritzi

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Found at google in 10 seconds:

http://www.uni-bonn.de/~manfear/mathdict.php

12.09.2008, 20:49

tigerbine

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