Finding the Inverse of an n x n Matrix Using Row Operations

hey I want to show you how you can use row operations to actually find the inverse of a matrix and the idea is kind of cool what you do is suppose you want to find the inverse of say this matrix this is a three by three matrix suppose you want to find the inverse of this matrix what you would first do is create an Augmented matrix that would be a three by six and in general if you're going to have an N by n matrix that you're trying to find the inverse of you're going to want to have n rows in the Augmented matrix and 2n columns because what you do is you take the a itself the matrix that you're trying to find the inverse of and then you augment it with the identity matrix which remember just has ones along the diagonal starting from the upper left down to the lower right and zeros everywhere else that's the identity matrix and now perform basic row operations again and again and again to transform this Augmented matrix which has a augmented with the identity to the identity matrix augmented with something else so the idea is to get all the ones along the diagonal and zeros everywhere else here using those basic row operations that we've talked about and then it turns out whatever is left here on the other side of this augmented matrix is in fact the identity is in fact the inverse for the original matrix a that's so cool so awesome and I want to show you an example with this example and so the first thing you do is you write an Augmented matrix where notice I copied a exactly and then here I augment it with the identity matrix so I put 1 0 0 0 1 0 and 0 0 1 ok awesome and now the goal is just to do row operations and to transform this side to look like this side so the idea here just to say it in kind of words is I want to get zeros all underneath here and then what I want to do is I want to get a 0 here and a zero here and then get a 0 here and a 0 here so and then just divide by whatever's there to make them all ones now the tricky thing is this kind of a little bit of an order to it you you don't want to do some of these steps kind of intermediately because you want to make sure first you get zeros all over here so that if you use things later you're not going to disrupt things over here it's a little tricky but we'll see it actually in practice so how are we going to start this so the way we're going to start this put this off to the side for a second so here's where we're going to start the story and now we're going to do row operations and so what do we do first well the first thing we're going to do is just keep that first row exactly as it is so notice that the first the first row I'm just going to keep exactly as it is just copy it down but now what I'm going to do I'm going to perform a row operation I'm going to replace the second row by the first row plus two times the second row and why well check it out if I multiply this row by two then this becomes a negative two and when I add it to the first row that's going to produce a zero and if we keep doing this now so I multiply this by two and I add it to this that's going to give me a six if I multiply this by two and add it to that that's going to be negative two plus one is going to be negative one and so forth and you continue down the pike and then there you have it so I just did this all the way through even on the other side of the dots of course great and then while I'm here I want to actually transform and change the the third row and I'm going to replace the third row by row one plus a negative two times and it should be Row three that's a little typo there it's okay it turns out you can easily make mistakes on these things by the way it is so easy to make mistakes trust me I know but that's a three because check it out you see if I multiply this by negative two and add it to this that becomes a zero and if I multiply this by negative two that becomes a negative eight and when I add four I get negative 4 and so forth down the pike so we go right through and just do that arithmetic and now we get an Augmented matrix where we've actually done a couple of of actual row row operations but notice that the important thing is I now have zeros here that's what I'm shooting for okay so let's keep going I don't need this one anymore I'm now going to just focus on this one and what am I going to do next well when we do next is I'm going to now try to work on getting a zero right here that's my next goal so how am I going to do that well the way I'm going to do that turns out is going to be to just add the third row to the first row now notice how cool this is when I add zero to two I'm still going to get that two so nothing's gonna change there and then when I add these two things notice they add to give zero and so there's the zero that I was really after and then when I have to continue this process I have to now add this and this and so now I'm going to get a two up there and you continue this process through here add these two things I get a 2 add these two things I get a 0 add these 2 things I get negative 2 no problem ok great now what am I going to do next well the next thing I want to do here and by the way there is not like a next step it's anything that you want to do just in order to make sure your following row operations and do what kind of you want to do so don't get kind of hung up and saying this there is a next step this is just how I'm doing it but you can do it lots of different ways I'm now going to actually while I'm here replace Row 2 by adding Row 3 in Row 2 just add these two now why would I do that well it's going to have no effect here so that's good of course and that's the point of this and here the effect is it'll make that a 2 which is a little bit smaller but here's why I really like it when I add these two things I get a free zero so I actually could kind of sneak in there I get a free zero add these two things I get a 2 2 and this is negative 2 2 2 negative 2 and then finally what I want here of course is a 0 right there so how do I get that the way I'm going to do that is by doing nothing right now I just faked you out but this is where we are right now so we made some progress and you can see slowly this is getting to look like an identity matrix lots of zeros and the diagonal is still the diagonal so awesome alright so now where do we go from here what would be good good next step well there's lots of possibilities I guess the the note thing I noticed is that these are all even and these are all even so just why like I think about it let me just clean those up and how do you get rid of all those things I can certainly divide or multiply through by by 2 or divide by 2 multiply through by half or / to whatever you want to say so let's just take half of r1 and if I take half of r1 of course everything just becomes 1 0 1 1 0 negative 1 similarly here I could take half of r2 and put that in place of the new r2 that's totally allowed 0 1 0 1 1 negative 1 0 1 0 1 1 negative 1 great and now I really want to get that 0 there so how do I do that well check out what I'm going to do I'm going to take 2 times the second row and add it to the third row 2 times 2 is 4 and when I add it to the negative 4 I get 0 and that's exactly what I want and notice that by doing that nothing is disruptive here because everything in sites is 0 0 times 2 is 0 plus 0 is 0 so we're good to go and now 2 times 2 is 4 plus negative 4 is 0 that's exactly what I want and then 2 times 0 is is 0 plus 1 is 1 and we keep doing this 2 times this is 4 + 1 is 5 2 times this is 4 plus 1 is is 4 as plus 0 is 4 and 2 times this is negative 4 minus 2 is negative 6 so now look how close we are this is really it's amazing we got here this fast this almost looks like the identity matrix the only thing left is to get rid of that one and notice that the way to do that is to take this and subtract that and notice that nothing else gets affected 1 minus 0 will still be 1 0 minus 0 will still be 0 but now we're going to have what we want so that's going to be the very last step which I'll to show to you we take our 1 and subtract our 3 or negative 3 negative r3 plus r1 and notice that when we subtract this minus that is 1 this minus that is 0 this minus that is 0 this minus that is negative 4 this minus that is negative 4 and this negative 1 minus negative 6 is 6 minus 1 which is 5 great so notice that this is the identity matrix which means that what's on this side in fact has to be the inverse of the original matrix that we started with really cool just using row operations and being really really careful with all the the algebra all the arithmetic anyway enjoy thinking about this interesting way of using row operations to find the inverse of a matrix and I'll see you soon

Channel: ThinkwellVids

Views: 318,230

Rating: 4.8045526 out of 5

Keywords: Thinkwell, Edward Burger, Math, Burger, Edward, Instruction, Ed, Algebra, College, textbook, mathematics, Matrices, Determinants, Inverses, n x n, Row Operations, Finding the Inverse of an n x n Matrix Using Row Operations

Id: HwRRdG_E4Yo

Length: 9min 48sec (588 seconds)

Published: Thu Mar 06 2014