Gaussian elimination, Mathematics

Assignment Help:

Example1:  Solve the subsequent system of equations.

-2x1 + x2 - x3 = 4

x1 + 2x2 + 3x3  = 13

3x1 + x3 = -1

Solution

The initial step is to write down the augmented matrix for above system. Keep in mind that coefficients of terms which aren't present are zero.

1471_Gaussian Elimination.png

Here, we want the entries below the main diagonal to be zero. The most main diagonal has been colored red thus we can keep track of it throughout this first illustration.  For reasons which will be apparent eventually we would prefer to find the main diagonal entries to all be ones suitably.

We can find a one in the upper most spot through noticing that if we interchange the first and second row we will find a one in the uppermost spot for free.  Therefore let's do that.

293_Gaussian Elimination1.png

This time we need to find the last two entries as -2 and 3 in the first column to be zero.  We can do this by using the third row operation. Note that if we get 2 times the first row and add this to the second row we will find a zero in the second entry into the first column and if we get -3 times the first row to the third row we will find the 3 to be a zero. We can do both of such operations at similar time so let's do that.

949_Gaussian Elimination2.png

Before proceeding along with the subsequent step, let's ensure that you followed what we just did. Let's see the first operation which we performed. This operation needs to multiply an entry in row 1 with 2 and add it to the consequent entry in row 2 after that replace the old entry in row 2 along with this new entry. The subsequent are the four individual operations which we performed to do this.

2 (1) + (-2) = 0

2 (2) + 1 = 5

2 (3) + (-1) = 5

2 (13) + 4 = 30

 

 

Okay, the subsequent step optional, although again is convenient to do. Technically, the 5th element in the second column is okay to leave. Conversely, it will create our life easier down the road if this is a 1. We can utilize the second row operation to support this. We can divide the entire row with 5. Doing it gives,

1023_Gaussian Elimination3.png

The subsequent step is to then utilize the third row operation to create the -6 in the second column in a zero.

1099_Gaussian Elimination4.png

Here, officially we are complete, but again it's somewhat convenient to find all ones on the main diagonal thus we'll do one last step.

1312_Gaussian Elimination5.png

We can now change back to equations.

2028_Gaussian Elimination6.png

     x1 + 2x2 + 3x3 = 13

⇒              x2 + x3 = 6

                   x3 = 2

At this point the solving is fairly easy.  We find x3 for free and once we find that we can plug it in the second equation and find x2. We can after that use the first equation to find x1. Remember as well that having 1's along the main diagonal helped somewhat along with this process.

The solution to that system of equation is,

x1 = -1

 x2  = 4

 x3  = 2

The process used in this example is termed as Gaussian Elimination.


Related Discussions:- Gaussian elimination

An even function, Assume that   i)  Determine all the roots of f...

Assume that   i)  Determine all the roots of f(x) = 0. ii)  Determine the value of k that makes h continuous at x = 3. iii)  Using the value of k found in (ii), sh

Statistics, How do you calculate for the distance between two co-ordinates?...

How do you calculate for the distance between two co-ordinates?

Define euler circuit and euler path, Define Euler Circuit and Euler Path.  ...

Define Euler Circuit and Euler Path.  Which of the following graphs have an Euler circuit and Euler path.

Possible outcome of a coin - probability based question, A coin is tossed t...

A coin is tossed twice and the four possible outcomes are assumed to be equally likely. If A is the event,  both head and tail have appeared , and B be the event at most one tail i

Rebecca is 12.5% taller than debbie how tall is rebecca, Rebecca is 12.5% t...

Rebecca is 12.5% taller than Debbie. Debbie is 64 inches tall. How tall is Rebecca? Because Rebecca is 12.5% taller than Debbie, she is 112.5% of Debbie's height (100% + 12.5%

What is a function, What is a Function, Anyway? Domain? Range? Next tim...

What is a Function, Anyway? Domain? Range? Next time you're at a fast-food restaurant, take a look at the price list. It may look something like this: • Hamburger.............

Math134, how to sketch feasible set

how to sketch feasible set

Levels of significance - rejection and acceptance regions, Levels of signif...

Levels of significance A level of significance is a probability value which is utilized when conducting tests of hypothesis. A level of significance is mostly the probability

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd