Gauss-Jordan Elimination

okay so the gauss-jordan method of solving a system of equations which have many many variables many many equations it's a great systematic way of actually finding solutions let's take a look at three equations three unknown an example and see if we could actually do this for ourselves so here are the three equations so X plus y plus 2z equals negative 1 X plus 3y minus 6 Z equals 7 and 2x minus y plus 2z equals 0 now we're going to follow the same systematic approach that is given by the gauss-jordan method that's the idea so the first thing I'm going to do is create this Augmented matrix the Augmented matrix is just given to us by literally considering the coefficients and writing them in the appropriate spots the X is the wise and the Z's and then we augment the constants here on the far right so we see for example reading down 1 1 2 1 3 negative 1/2 negative 6 2 negative 1 7 0 and voila like a cooking show I've done this in advance and there it is so this is our augmented matrix member I've augmented the constants so in fact the variables and the addition all gone away we understand this is the X column this is the Y column and this is the Z column and that's it needs 2 constants great so in fact this is now served its purpose we can go here what's our goal our goal is to massage this matrix using the method of Gauss Jordan and convert it into an equivalent matrix that would look like this now why is that good well this is great because you see this tells me that 1 times X plus 0 times y plus 0 times Z equals a so namely x equals a that's what I get out of here and then similarly here I'd see that 0 times X plus 1 times y plus 0 times Z equals B namely y equals B and similarly Z equals C so if I can transform this matrix into a form of this kind then in fact I can just literally read off the answers and what are the rules the rules are that I can multiply a matrix by a number and add it to another matrix and put it in its stead and so forth and that's the idea so let's see if we can shoot to this that's the goal all right let's roll up our sleeves ok enough dilly-dallying begin so the first thing I want to do if you recall is to make this a zero and make this a zero so I just have one zero zero so how do I do that well in this case what I want to do let's see how I'm going to do this you have any ideas I don't what I want to do is I'm going to multiply this column with this row rather by negative one and then I'm going to add it to this row now why is that a good idea if I multiply this r1 let's call it by negative one and then add it to this then I see negative one and one is zero and that's all I'm shooting for here is to get a zero and then I'll do whatever else follow suit in order to make this kind of go so this is going to be transformed into a new Augmented matrix that's equivalent but different I'm going to keep the top row as it is and now what I'm going to do I'm going to take I'm going to take this row and add it to negative one times this row so this minus this is 0 now I just have to follow along and put in whatever else I need to put in so 3 minus 1 is 2 and negative 6 minus a minus minus minus 2 is negative 8 I'm sorry now is negative 8 hmm everyone happy and then 7 minus negative 1 is going to be 7 plus 1 which is 8 great okay I'm happy I'm happy anything you're happy to okay now what we want to do is we want to actually produce a 0 here so how are we going to produce a 0 here what we're going to do is we're going to take this row and now multiply by 2 in fact negative 2 so we multiply this by negative 2 and then we add it to this because if I multiply this by negative 2 that makes this a negative 2 and 2 would be 0 remember the goal is to get a 0 here so multiply this by negative 2 so I get a 0 here then this is negative 2 so I have negative 3 and negative 2 is negative I'm sorry negative 1 and negative 2 is negative 3 I'm getting ahead of myself and then here I see a 2 and then I'm subtracting 2 times 2 which is 4 so 2 minus 4 which is negative 2 and here I see a 0 times 0 minus a negative 2 times negative 1 which is just a 2 and so I get this so in the first move notice I've done exactly what I wanted namely I got zeros below this one that's great now what would I'd like to do what I'd like to do now is I'd like to get a 0 below this 2 so this now is is done for the moment and now we're going to try and see if we can do more magic so how am I going to get to the next equivalent augmented matrix well let's see if I can do this ok so now what I want to do is I want to make this 0 so I should multiply this by something in which I multiply it by well it seems like three halves because if I multiply this by three halves then this would actually just become a 3 and when I add it to negative 3 I get 0 so in fact I'm not going to do anything here but I'm going to multiply everything there by 3 halves so if I multiply this by 3 halves I get 0 and when I add it to this I get 0 which of course should always be the case we shouldn't undo all the great work we've been doing multiply this by 3 halves that this gives me a 3 when I add this negative 3 I get 0 multiply this by 3 halves so that's actually going to be negative 4 times 3 negative 4 times 3 is negative 12 and then I add to add negative 2 I get negative 14 and I multiply this by three halves and multiply this by 3 halves I just get 12 and I add 12 to 2 2 and I get 14 so there I have it so that's where I am now and notice that now it's a triangular matrix you still with me I hope so I'm barely with me so here we go now we can actually simplify in this case I'm going to actually elect to do that since the other so many numbers running around here because notice I could take this equation and actually divide it by two so if I take this equation and divide it by two then what happens then I get a zero here and then here I get a one and then here I get a negative four and here I get a four so I already get a one here which is absolutely awesome now what I do here actually if I divide that by negative 14 I would see zero zero divided by negative 14 is zero zero divided by negative 14 is still zero negative 14 divided by negative 14 is one and 14 divided by negative 14 is negative one so I get an Augmented matrix that looks like this which is still equivalent to the old one so this is great I'm actually I'm making progress because now I've got ones along the diagonal what do I have to do I hope to get zeros up on top so how am I going to get zeros up on top okay well now we've got to take a look at what we can do well now we use the bottom the bottom roast in order to get the zero so for example what I should do here is take this row and subtract this row so take Row one and then add negative 1 times Row 2 because that will give me a 1 minus 1 which is 0 so if I take this and subtract that I get 1 if I take this and subtract that I get 0 if I take this and subtract that that's going to be actually 2 minus negative 4 which is going to be a 6 and if I take this and subtract that I'm going to get negative 5 augmented negative 5 so there so now I've got a 0 there which is awesome in fact I would I'll do is I'll keep everything else here just for the moment because I want to take this in step so you can really see everything unfolding this is an equivalent system to this matrix ok now what do I want well now what I want to do is I want to get make these things 0 I'm getting close I'm getting close here we go now this is going to be a little tricky let's see if we can figure this out okay how do I make that zero well I shouldn't use this thing here because then I'll get this one knotted up in here and I'll lose that zero so I got to go here this means that I won't change anything in the first two and now I'm going to deal with this how I'll multiply this by negative six and then I'll add it to Row one so I'm going to take Row one plus negative six times Row three and put that in the new Row one so I'm going to basically do this I'm going to take this minus six times that this minus six times that is just one this minus six times that is 0 this minus six times this is happily 0 which was the whole point this minus 6 times this that's going to be negative 5 minus negative 1 times 6 which is going to be 6 so negative 5 plus 6 is 1 okay now I'm actually gonna do the other step as well I'm going to make this a zero at the same moment what do I do well here to make that a zero I should multiply this by 4 and just add it to this so the operation I'm going to do here is to replace Row two by Rho 2 plus 4 times Row three and when I do that what do I see so I'm going to multiply this by 4 and add so 0 plus 0 times 4 is 0 1 plus 0 times 4 is still 1 negative 4 plus 4 times 1 is 4 minus 4 which is 0 great and then I have 4 minus a 1 times 4 which would be 4 minus 4 to 0 okay and the last one is all set look at us wow I think I survived this to you and I get to this Augmented matrix now notice this is exactly exactly of the form we desire because now we can just read off the answer we could say this tells me this was the X remember way back long ago in olden days we had this equation so the first the first column are the X's so x equals 1 I equals zero and Z equals negative 1 well now after all of that arithmetic and knowing how bad I am at actually computing things you might say is this correct well let's actually check our answer so we can check our answer really easily by inserting this value for X 1 and y 0 and Z negative 1 and see if we get the same answers and we have 2 by the way check it for each and every one of the three original equations so in place of X will be 1 in place of Y we'll have 0 and in place of Z we'll have negative 1 let's go first equation I see 1 plus 0 minus 2 well that's 1 minus 2 which is negative 1 great next I see 1 plus 0 minus 6 times negative 1 that's plus 6 so I have plus 6 plus 0 plus 1 that's 7 and finally big finish let's have a drumroll thank you I have 2 times 1 minus 0 plus 2 times negative 1 so that's 2 minus 0 minus 2 so 2 minus 2 is wow I did this flawlessly which by the way good for me good for me the point is that you can do it as well as long as we're careful if you have a very very large system in fact computers will actually able to do this with great ease but it's wonderful to see for ourselves that we can start with an Augmented matrix and then after a little bit of toiling and just using some basic arithmetic but using the power of the Gauss Jordan method we can actually resolve and find x y&z for ourselves on the deserted island on that beach in congratulations Isis

Channel: ThinkwellVids

Views: 331,666

Rating: 4.8046627 out of 5

Keywords: Thinkwell, Edward Burger, Math, Burger, Edward, Instruction, Ed, Algebra, College, textbook, mathematics, Matrices, Determinants, Systems, Equations, Gauss-Jordan, Elimination

Id: 0fTSBIBD7Cs

Length: 12min 59sec (779 seconds)

Published: Wed Mar 05 2014