College Algebra: Using the Gauss-Jordan Method

Gauss-Jordon

10/21/2009

~ Assessment1

Great clear explanation of the Gauss-Jordan elimination. Before watching this video I was under the miss-conception that there was a particular ORDER to solving to achieve the 1's and 0's,meaning I though we must solve m11, then m21 in that order. Nothing was mentioned and therefore I've learned how to much more simply apply the rules of the Gauss-Jordan elimination technique.
When you watch, be sure and pause, take note of the rules. Great!

Helped a lot!

10/01/2009

~ Christopher5

While this is sort of a basic overview of Glauss-Jordan methodology, he really explained it about 1000 times better in seven minutes than my Finite Math teacher did in two hours. I wish I had this teacher for my class. :(

gaussman-jordan method

07/27/2009

~ maria3

very well explained

College Algebra: Using the Gauss-Jordan Method

03/25/2009

~ rhonda1

Helpful video to understanding the Gauss-Jordan method.

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I first want to take a look at these matrices in terms of solving these systems of equations. That was sort of the first inspiration for this collection of numbers. So, consider this system right here. Here we have x + y = 5 and x - y = -1. Now, certainly there are a variety of ways of solving this simple system. You could use the elimination method. You can substitute and so forth, but now I want to try to use this simple example as a template to the idea of using these matrices technique. In fact, this technique for solving this system is called the Gauss-Jordan Method. Here’s what the method is. The method is let’s just make a pact right now that here will always be x’s and here will always be y’s and here will always be constants.
If I make that pact with you, then I can actually forget about those variables and just write this instead of the first equation. Because there’s a one in the x spot, a one in the y spot and a five over here. Do you see how these two things are actually the same? What would I write here? Well, here I’d write a one and a minus one and a minus one. So, I can strip away the equal signs and all the other stuff, the x’s and the y’s, as long as you promise me x’s go here and y’s go here. Okay, now what can I do? If you think of this now as a matrix? If you think of this now as a collection of just numbers in a grid like this? This would be a two by three grid and it’s something that’s called a augmented matrix because I have this little dotted line here to remind me these are variable x, y and these are the numbers.
So, there’s a little matrix right there. Now, what things could I do to this that would be legal operations in terms of the legal operations I could do with these equations? Well, if you just think about what you can do with equations, you can immediately see what those rules should be here in the Gauss-Jordan method. Rule number one is I can, first of all, flip the order of these. By doing this it’s still the same system. That would be equivalent to doing this. So, flipping a row with another row is no problem, because it’s just basically moving around the equations. So, that’s fine.
What else could I do? Well, I can always take an equation and multiply it through by something. That doesn’t change the equation. So, I can also do that here. I could take any row I want and multiply that row through by a constant. Like I can multiply everything here by three if I wanted to, 3, 3, 15. That would be equivalent to multiplying this equation through by three, 3x + 3y = 15. It doesn’t change the equation. So, I could actually multiply through by something, as long as I go all the way through. What else do I tend to do with these equations? When I do the substitution I like to be able to multiply an equation through by something and then add it to another equation. Well, I can do that here too. What would that look like?
I could take any row and actually multiply it through by something and add it to another row and replace that other row by that addition. So, that would actually be multiplying through by three and then adding it to this row and replacing that row by what the answer is. That’s basically taking this, multiplying it by three and adding it to this equation and putting that equation right there. It doesn’t change the value. So, those are the basic rules of the Gauss-Jordan Method. You can sort of flip the order of these things. That’s not a big deal at all. You can multiply any row through by a constant and you can replace any row by taking another row, multiplying through by something and then adding it or subtracting it to that row.
Now, just using those methods we can actually now solve this system just using this matrix. Now, what’s the goal? Well, the goal at the end of the day would be to have a matrix that looks like this. Let me just draw you what the goal would look like. So, here’s our fantasy. At the end of the day we want one, zero, zero, one, dot, dot, dot, dot, dot, dot and then some number here and some number here. Let’s see why that really is our goal. That’s our goal because if we have that what does that mean? What’s the top equation?
One x and no y’s equals that. So, that means x equals that number. So, that means x equals. What would this mean? No x’s and one y. So, y equals that. So, if I could get this system to be in this form that would mean that I know what x equals and I know what y equals. I can just read them off the right hand column. So, my goal is to take those rules I just told you about, switching rows, subtracting rows and adding them and so forth, and to try to massage this to look like that. That would actually be solving this system of equations.
So, let’s actually try to do that right now. So, what I do is I’m going to start with this system right here. I'm going to write that up here. I have one, one, one, minus one and I have five and a minus one. So, there’s the system. Remember this is always the x’s and these are always the y’s. Now, if I dispense with this, my goal is to get this to look like this using those legal operations. Which really just convert down to operations with equations. So, the first thing I’ll do is I’ll actually take this row and add it to that row and put that in this row. So, I’m going to replace this row by the sum of this row and that row. Why? Because if I add these two things that will put a zero right there and that’s a good thing. I want a zero there.
So, if I do that what I’m doing here is I’m going to, in this row I’m going to take row one and add row two. So, in place of this I’m going to put the sum of these two rows. So, just add, one and one is two. One and minus one is zero. Five and minus one is four. I’ll do nothing here. That’s the equivalent, basically, of adding the two equations up and this is what you get. Now, I want a one here, so I’ll just divide the top thing through by two. Which is a legal move, because that’s just taking the equation and dividing it through by two. So, when I do that I get one, zero, two. I still have this, one, minus one, minus one, but now what could I do?
Well, I could take this and subtract this and put that in here. That's just the substitution, which would look like this. What I’m going to do is right here I’m going to put in, let’s see, I’m going to take row one and subtract off row two. So, this minus--let me put this, first of all, I’m not going to do anything up here. So, this minus that is zero. This minus that becomes a one. This minus that becomes a three. Look, I got it in my fantasy form. If you remember this is the x and this is the y, this means x = 2. So, x = 2 and this means y = 3.
So, now if I go back to the original equation I should have this thing satisfied. If x = 2 and y = 3, their sum is 5. Yet if I take two and subtract three I get negative one. This actually works. Notice what I did. I just did all the manipulation involved in looking at the elimination method, but instead of doing it with all the x’s and y’s there, I just use the coefficients and carefully moved things around by adding rows to other rows and so forth and getting it into this form. This is the Gauss-Jordan Method. You want ones along this diagonal here and zeros everywhere else, so you can just say x equals, y equals. Okay, we’ll take a look at this method again up next.