# Example of Gauss-Jordan elimination algorithm

## Example of Gauss-Jordan elimination algorithm

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 (-pē+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 (pē-2p-3) : 0 ]

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

pē - 2p - 3 = 0

Δ = bē - 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) / (pē-2p-3)

- R(2) / (-pē+2p+3)

- R(3) / (pē-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.

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

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 (-pē+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 (pē-2p-3) : 0 ]

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

pē - 2p - 3 = 0

Δ = bē - 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) / (pē-2p-3)

- R(2) / (-pē+2p+3)

- R(3) / (pē-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.

_________________

The Cosmos is full beyond measure of elegant truths

Here I come to save the daaaaaaay!

Counciller-Builder of Cold

**Storm**- Councillor
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## Re: Example of Gauss-Jordan elimination algorithm

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

_________________

The Cosmos is full beyond measure of elegant truths

Here I come to save the daaaaaaay!

Counciller-Builder of Cold

## Re: Example of Gauss-Jordan elimination algorithm

What the f u c k is that?

**Huni**- Member
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## Re: Example of Gauss-Jordan elimination algorithm

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...

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...

_________________

*My hand is shaking... or is my whole body?*

My heart is beating furiously, my breath betrays me...

I can't control my feelings now. The pain turns to anger and back again.

I may gain strength through my passion, but feel weak the same time.

Myself is changing, don't know who I am...

Oh my God, what have I become...?

My heart is beating furiously, my breath betrays me...

I can't control my feelings now. The pain turns to anger and back again.

I may gain strength through my passion, but feel weak the same time.

Myself is changing, don't know who I am...

Oh my God, what have I become...?

**C.O.L.I.N.**- Master
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Age : 31

Location : Athens, Greece

Registration date : 2007-12-06

## Re: Example of Gauss-Jordan elimination algorithm

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

_________________

The Cosmos is full beyond measure of elegant truths

Here I come to save the daaaaaaay!

Counciller-Builder of Cold

## Re: Example of Gauss-Jordan elimination algorithm

i think we simply call it "3x3 system" btw (3 parameters in 3 equations)

i odnt remember much of methods names anyway

i odnt remember much of methods names anyway

_________________

*My hand is shaking... or is my whole body?*

My heart is beating furiously, my breath betrays me...

I can't control my feelings now. The pain turns to anger and back again.

I may gain strength through my passion, but feel weak the same time.

Myself is changing, don't know who I am...

Oh my God, what have I become...?

My heart is beating furiously, my breath betrays me...

I can't control my feelings now. The pain turns to anger and back again.

I may gain strength through my passion, but feel weak the same time.

Myself is changing, don't know who I am...

Oh my God, what have I become...?

**C.O.L.I.N.**- Master
- Number of posts : 6677

Age : 31

Location : Athens, Greece

Registration date : 2007-12-06

## Re: Example of Gauss-Jordan elimination algorithm

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

_________________

The Cosmos is full beyond measure of elegant truths

Here I come to save the daaaaaaay!

Counciller-Builder of Cold

## Re: Example of Gauss-Jordan elimination algorithm

You must be bored.

**Kung-Pow**- Admin
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## Re: Example of Gauss-Jordan elimination algorithm

This is no entertainment.

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*You are not an adult before you dare being childish"*

***Hule***- Master
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## Re: Example of Gauss-Jordan elimination algorithm

Lol.. K I agree with you. =P*Hule* wrote:This is no entertainment.

But it is entertaining to boom. xD

**Tony***- I have no life
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## Re: Example of Gauss-Jordan elimination algorithm

That's unnatural. Absolutely abnormal. Completely perverted.

_________________

"

*You are not an adult before you dare being childish"*

***Hule***- Master
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Age : 26

Location : Kakariko Village.

Registration date : 2007-12-02

## Re: Example of Gauss-Jordan elimination algorithm

Don't ask why I post this, I just searched mathematics on google cause I was bored.

**Tony***- I have no life
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Registration date : 2008-07-01

## Re: Example of Gauss-Jordan elimination algorithm

I find these things to be entertaining to do.

_________________

The Cosmos is full beyond measure of elegant truths

Here I come to save the daaaaaaay!

Counciller-Builder of Cold

## Re: Example of Gauss-Jordan elimination algorithm

lol

actually in greek sounds reasonable

like many terms of mathematics and medical terminology (and in all the other sciences that have their routes form back then)

actually in greek sounds reasonable

like many terms of mathematics and medical terminology (and in all the other sciences that have their routes form back then)

_________________

*My hand is shaking... or is my whole body?*

My heart is beating furiously, my breath betrays me...

I can't control my feelings now. The pain turns to anger and back again.

I may gain strength through my passion, but feel weak the same time.

Myself is changing, don't know who I am...

Oh my God, what have I become...?

My heart is beating furiously, my breath betrays me...

I can't control my feelings now. The pain turns to anger and back again.

I may gain strength through my passion, but feel weak the same time.

Myself is changing, don't know who I am...

Oh my God, what have I become...?

**C.O.L.I.N.**- Master
- Number of posts : 6677

Age : 31

Location : Athens, Greece

Registration date : 2007-12-06

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