Example of Gauss-Jordan elimination algorithm

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Example of Gauss-Jordan elimination algorithm

Post by Boomlala on Thu Dec 04, 2008 11:26 am

Due to impopular demand, I'll give you a taste of my math skills.

=========================

We are given the following system of parametric linear equations:

{x + (p-1)y + z = 0
{x + y + (p-1)z = 0
{(p-2)x - y + 4z = 0

First we derive the extended matrix of this system:

[ 1 (p-1) 1 : 0 ]
[ 1 1 (p-1) : 0 ]
[ (p-2) -1 4 : 0 ]

operations performed:
- a(11)R(2) - a(21)R(1)
- a(11)R(3) - a(31)R(1)

[ 1 (p-1) 1 : 0 ]
[ 0 (-p+2) (p-2) : 0 ]
[ 0 (-p+3p-3) (-p+6) : 0 ]

We solve the equation: -p + 2 = 0

-p + 2 = 0
-p = -2
p = 2

------------
Case I : p = 2
------------

operations performed:
- substitution of p
- R(2) switched with R(3)
- a(22)R(1) - a(12)R(2)
- a(22)R(3) - a(32)R(2)

[ -1 0 -5 : 0 ]
[ 0 1 4 : 0 ]
[ 0 0 0 : 0 ]

We derive a new system of this extended matrix:

{-x - 5z = 0
{-y + 4z = 0
ergo
{x = -5z
{y = 4z
ergo
V(I) = {(-5z, 4z, z) | z ∈ ℝ}

--------------
Case II : p ≠ 2
--------------

operations performed:
- R(2) / (p-2)
- a(22)R(1) - a(12)R(2)
- a(22)R(3) - a(32)R(2)
- -R(1)

[ 1 0 p : 0 ]
[ 0 -1 1 : 0 ]
[ 0 0 (p-2p-3) : 0 ]

We solve the equation: p - 2p - 3 = 0

p - 2p - 3 = 0
Δ = b - 4ac = 16
sqrt(Δ) = 4 (sqrt(x) = square root of x)
V = {( (-b+sqrt(Δ))/(2a) , (-b-sqrt(Δ))/(2a) )}
V = {(3,-1)}

-----------------
Case IIa : p = 3
----------------

operations performed:
- substitution of p

[ 1 0 3 : 0 ]
[ 0 -1 1 : 0 ]
[ 0 0 0 : 0 ]

We derive a new system out of the extended matrix, like we did in case I:

{x + 3z = 0
{-y + z = 0
ergo
{x = -3z
{y = z
ergo
V(IIa) = {(-3z,z,z) | z ∈ ℝ}

------------------
Case IIb : p = -1
-----------------

operations performed:
- substitution of p

[ 1 0 -1 : 0 ]
[ 0 -1 1 : 0 ]
[ 0 0 0 : 0 ]

Once more, we derive a new system:

{x - z = 0
{-y + z = 0
ergo
{x = z
{y = z
ergo
V(IIb) = {(z,z,z) | z ∈ ℝ}

------------------
Case III : p ∉ {-1,3}
------------------

operations performed:
- a(33)R(1) - a(13)R(3)
- a(33)R(2) - a(23)R(3)
- R(1) / (p-2p-3)
- R(2) / (-p+2p+3)
- R(3) / (p-2p-3)

[ 1 0 0 : 0 ]
[ 0 1 0 : 0 ]
[ 0 0 1 : 0 ]

For the last time, we derive a new system

{x = 0
{y = 0
{z = 0
ergo
V(III) = {(0,0,0)}

----------------
Solution
----------------

Now, we take all our solution sets (V(I), V(IIa), V(IIb) and V(III)) and create one single solution set, V.

V =
{(-5z,4z,z) | z ∈ ℝ ; p = 2 }
{(-3z,z,z) | z ∈ ℝ ; p = 3 }
{(z,z,z) | z ∈ ℝ ; p = -1 }
{(0,0,0) | p ∈ ℝ \ {-1,2,3} }

Solved.

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Re: Example of Gauss-Jordan elimination algorithm

Post by Storm on Thu Dec 04, 2008 11:30 am

Thank you.

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Re: Example of Gauss-Jordan elimination algorithm

Post by Matthew on Thu Dec 04, 2008 11:32 am

Is this arrogance in its most subtle form?
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Re: Example of Gauss-Jordan elimination algorithm

Post by Boomlala on Thu Dec 04, 2008 11:33 am

Nay. Just an example of what I was talking 'bout in the 'Blog. And something I find very amusing to do.

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Re: Example of Gauss-Jordan elimination algorithm

Post by Huni on Thu Dec 04, 2008 11:51 am

What the f u c k is that?
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Re: Example of Gauss-Jordan elimination algorithm

Post by C.O.L.I.N. on Thu Dec 04, 2008 12:00 pm

i just understood (confirmed again actually) y most greeks have a problem with maths at the next years
at the age of 15-16 (4th grade of high-school) we did that stuff... i think we had 4 hours only of these mathematics per week and these mathematics r common for every student...
while in Booms school, they do them too, with way more hours and to students that want to have mathematics as one of their main subjects...
shame.. they should have moved slower here.. otherwise we would have learn more and better...

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Re: Example of Gauss-Jordan elimination algorithm

Post by Boomlala on Thu Dec 04, 2008 12:13 pm

h4x0l@tEr/Huni wrote:What the f u c k is that?

Solving this system of parametric linear equations:

{x + (p-1)y + z = 0
{x + y + (p-1)z = 0
{(p-2)x - y + 4z = 0

Basically finding the values of x, y and z which, when filled in, give correct statements.

EDIT: spelling mistake


Last edited by Boomlala on Thu Dec 04, 2008 12:36 pm; edited 1 time in total

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Re: Example of Gauss-Jordan elimination algorithm

Post by C.O.L.I.N. on Thu Dec 04, 2008 12:17 pm

i think we simply call it "3x3 system" btw (3 parameters in 3 equations)
i odnt remember much of methods names anyway Razz

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Re: Example of Gauss-Jordan elimination algorithm

Post by Boomlala on Thu Dec 04, 2008 12:36 pm

Well, there is only one parameter though, namely p.

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Re: Example of Gauss-Jordan elimination algorithm

Post by Kung-Pow on Thu Dec 04, 2008 12:38 pm

You must be bored.
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Re: Example of Gauss-Jordan elimination algorithm

Post by *Hule* on Thu Dec 04, 2008 2:18 pm

This is no entertainment.

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Re: Example of Gauss-Jordan elimination algorithm

Post by Tony* on Thu Dec 04, 2008 2:19 pm

*Hule* wrote:This is no entertainment.
Lol.. K I agree with you. =P
But it is entertaining to boom. xD
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Re: Example of Gauss-Jordan elimination algorithm

Post by *Hule* on Thu Dec 04, 2008 2:20 pm

That's unnatural. Absolutely abnormal. Completely perverted.

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Re: Example of Gauss-Jordan elimination algorithm

Post by Tony* on Thu Dec 04, 2008 2:23 pm

Don't ask why I post this, I just searched mathematics on google cause I was bored.
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Re: Example of Gauss-Jordan elimination algorithm

Post by Boomlala on Fri Dec 05, 2008 4:43 am

I find these things to be entertaining to do.

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Re: Example of Gauss-Jordan elimination algorithm

Post by C.O.L.I.N. on Fri Dec 05, 2008 8:51 am

lol Razz
actually in greek sounds reasonable Wink
like many terms of mathematics and medical terminology (and in all the other sciences that have their routes form back then)

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