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# Deriving a method for determining inverses

## Video transcript

I have this matrix a here that I want to put into reduced row echelon form and we've done this multiple times you just perform a bunch of row operations but what I want to show you in this video is that those row operations are equivalent to linear transformations on the column vectors of a so let me show you by example so if we just want to put a into reduced row echelon form the first step that we might want to do if we want to zero out these entries right here is let me do it right here is will keep our first entry the same so for each of these column vectors we're going to keep the first entry the same so there's going to be one minus one minus one and actually let me simultaneously construct my transformation so I'm saying that what my role I'm going to perform is equivalent to a linear transformation on the column vector so it's going to be a transformation that's going to take some column vector a 1 a 2 and a 3 just going to take each of these and then do something to them and do something to them in a linear way there'll be linear transformations so we're keeping the first entry of our column vector the same so this is just going to be a 1 it's just going to be a 1 this is a line right here that's going to be a 1 now what can we do if we want to get to reduced row echelon form we'd want to put make this equal to a 0 so 1 B we want to replace our second row with the second row plus the first row because then these would guys would turn out to be 0 so let's let me write that in my transformation I'm going to replace the second row with the second row plus the first row with the second row plus the first row and let me write it out here minus 1 plus 1 is 0 to plus minus 1 is 1 3 plus minus 1 is 2 now we also want to get a 0 here so let me replace my third row with my third row minus my first row so I'm going to replace my third row with my third row minus my first row so 1 minus 1 is 0 1 minus minus 1 is 2 4 minus minus 1 is 5 just like that so you see this was just a linear transformation in any linear transformation you could actually represent as a matrix vector product so for example this transformation I could represent it and let me to figure out its transformation matrix so if we say that T of X is equal to I don't know let's call it some let's call it some at some matrix s times X we already used the matrix a so I have to pick another letter so how do we find s well we just apply the transformation to all of the column vectors or the standard basis vectors of the identity matrix so let's do that so the identity matrix I'll draw it really small like this the identity matrix looks like this 1 0 0 0 1 0 0 0 1 that's what the identity matrix looks like to find the transformation matrix we just apply this guy to each of the column vectors of this so what do we get do a little bit bigger so the first we apply it to each of these column vectors but we see the first row always stays the same so the first row is always going to be the same thing so 1 0 0 I'm essentially applying it simultaneously to each of these column vectors saying look when you take each of when you transform each of these column vectors their first entry stays the same the second entry the second entry becomes the second entry plus the first entry so 0 plus 1 is 1 1 plus 0 is 1 0 plus 0 is 0 and then the third entry the third entry gets replaced with the third entry minus the first entry so 0 minus 1 is minus 1 0 minus 0 is 0 and then 1 minus 0 is 1 now notice when I apply this transformation to the column vectors of our identity matrix I essentially just perform those same row operations that I did up there I perform those exact same row operations on this identity matrix but we know that this is actually the transformation matrix that if we multiply it by each of these column vectors we will or by each of these column vectors we're going to get these column vectors so you could view it this way let's let me call this this is right here this is equal to s this is our transformation matrix so if we could say that we could say that s if we create a new matrix whose columns are s times this column vector s times 1 minus 1 1 and then the next column is s times s times I wanted to do in that other color s times this guy minus 1 2 1 and then the third column is going to be the third column is going to be s times s times this third column vector minus 1 3 for this product we now know we're applying this transformation this is s times each of these column vectors that is the matrix representation we will this guy right here this guy will be transformed to this right here that'll become let me do it down let me do it oh down here this guy I wanted to show that stuff that I had above here as well well I'll just draw an arrow that's probably the simplest thing this matrix right here will become that matrix right there so another way you could write it this is equivalent to what what is this equivalent to when you take a matrix and you multiply it times each of the column vectors when you transform each of the column vectors by this matrix this is the definition of a matrix matrix product this is equal to our matrix s I'll do it in pink this is equal to our matrix s which is 1 0 0 1 1 0 minus 1 0 1 times our matrix a times times 1 minus 1 1 minus 1 2 1 minus 3 sorry minus 1 3 4 so let me make this very clear this is our matrix this is our transformation matrix s this is our matrix a and when you perform this product when you perform this product you're going to get this guy right over here you're going to get this guy right over here just copy and paste it edit let me copy let me paste it you're going to get that guy just like that now the whole reason why I'm doing that is just to remind you that when we perform each of these row operations we're just multiplying we're performing a linear transformation on each of these columns and it is completely equivalent to just multiplying this guy by some matrix s in this case we took the trouble of figuring out what that matrix s is but any of these row operations that we've been doing you can always represent them by a matrix multiplication you can always represent them by a matrix multiplication so this leads to a very interesting idea this leads to a very interesting idea when you when you put something in reduced row echelon form let me do it up here so let me let me actually let's just finish what we started with this guy let's put this guy in a reduced row echelon form so this we already said this is equal to let me call this first s let's call that s1 so this guy right here is equal to that first s-1 times a we already show that that's true now let's perform another transformation or let's just do another set of row operations to get us to reduced row echelon form so let's keep our middle row the same 0 1 2 and let's replace the first row with the first row plus the second row because I want to make this a 0 so 1 plus 0 is 1 let me do it in another color 1 plus 0 is 1 minus 1 plus 1 is 0 minus 1 plus 2 is 1 now I want to replace the third I want to replace the third row what let's say let's say the third row minus 2 times the first row third row minus 2 times the first row so that's 0 minus 2 times 0 is 0 2 minus 2 times 1 is 0 5 minus 2 times 2 is 1 5 minus 4 is 1 and we're almost there we just have to zero out these guys right there see if we can get this into reduced row echelon form so what is this I just performed another linear transformation actually let me write this let's say if this was a 4 linear transformation what I just did is I performed another linear transformation T 2 I'll write in a different notation where you give me some vector some column vector X 1 X 2 X 3 what did I just do what was the transformation that I just performed my new vector I made the top row equal to the top row plus the second row so that's X 1 plus X 2 I kept the second row the same and then the third row I replaced it with the third row minus 2 times the second row that was a linear transformation we just did and we could represent this linear transformation as being we could say t2 applied to some vector X is equal to some transformation vector s 2 times our vector X now we could say that this this is equal to we could say because if we applied this transformation matrix to each of these columns it's equivalent to multiplying this guy by this transformation matrix so you could say that this guy right here if we haven't figured out what this is but I think you get the idea this matrix right here is going to be equal to this guy it's going to be equal to s 2 times this guy and what is this guy right here well this guy is equal to s 1 times a this is going to be s 2 times s 1 times a fair enough so this is just s 2 times s 1 times a and you could have gotten straight here if you created a new if you just multiply it s 2 times s 1 this could be some other matrix and you're just multiplied it by a you'd go straight from there to there fair enough now we still haven't gotten this guy in reduced row echelon form so let's try to get there and I've run out of space below him so I'm going to have to go up so let's go upwards let's go upwards like this and what I want to do is I'm going to keep the third row the same keep the third row the same 0 0 1 and let me replace let me replace the second row with the second row minus 2 times the third row so we'll get a 0 we get a 1 minus 2 times 0 and we get a 2 minus 2 times 1 so that's a 0 and let's replace the first row with the first row - the third row so 1 minus 0 is 1 0 minus 0 is 0 and 1 minus 1 is 0 just like that and since we did them for the ellipsis actually right what our transformation was let's call it t3 I'll do it in purple t3 t3 is the transformation on some vector X let me write it like this on some vector X 1 X 2 X 3 it was equal to what did we do we replace the first row with the first row minus the third row x1 minus x3 we replaced the second row with the second row minus 2 x the third row so it's x2 minus 2 times x3 and then the third row just stayed the same so obviously this could also be represented this could also be representation matrix s 3 times X so this transformation when you multiply it to each of these columns is equivalent to is equivalent to multiplying this guy times this transformation matrix which we haven't found yet well we can write it so this is going to be equal to s 3 s 3 times this matrix right here which is s 2 s 2 s 1 a and what do we have here we got the identity matrix we put it in reduced row echelon form we got the identity matrix and we already know from previous videos if you the reduced row echelon form of something is the identity matrix then we are dealing with an invertible transformation or an invertible matrix because this obviously could be the transformation for some transformation let's just call this transformation I don't know let's call it T did I already use T buses call it T not or our transformation applied to some vector X that might be equal to ax so we know that this is we know that this is invertible because we put it in reduced row echelon form we put its transformation matrix in reduced row echelon form and we got the identity matrix so that tells us this is invertible but something even more interesting happened we got here by performing some row operations and we said those row operations or equivalent or completely equivalent to multiplying this guy right here by getting multiplying our original transformation matrices matrix by a series of transformation matrices that represent our row operations and when we multiplied all this this was equal to the identity matrix now in the last video we said that the inverse matrix so if this is T naught T naught inverse T naught inverse could be represented it's also linear transformation it can be represented by some inverse matrix that we just called a inverse times X and we saw that a inverse a inverse times or the inverse transformation matrix times our transformation matrix is equal to the identity matrix we saw this last time we prove this to you now something very interesting here we have a series of matrix products times this guy times this guy that also got me the identity matrix so this guy right here the series of matrix products this must be this must be the same thing as my inverse matrix as my inverse transformation matrix and so we could actually calculate if we wanted to we could actually just like we did we've actually figured out what s1 was we did it down here we could do a similar operation to figure out what s2 was s3 was and then multiply them all out and we would have actually constructed a inverse but something I guess something more interesting we could do instead of doing that what if we started off what if we applied the same if we applied the same matrix products or to the to the identity matrix so the whole time we did here when we did our first row operation so we have here we have the matrix a and let's say we have an identity matrix on the right let's call that I right there now our first linear transformation we did we saw that right here that was equivalent to multiplying s1 times a right the first set of row operations was this it got us here now if we perform that same set of row operations on the matrix what are we going to get we're going to get the matrix s1 s1 times the identity matrix is just s1 all of the columns of anything times the identity times the standard basis columns will just be equal to themself and you'll just be left with that s1 or you can call this as s1 times I that's just s1 fair enough now you perform your next row operation and you ended up with s2 times s1 times a now if you put perform that same row operation on this guy right there what would you have you would have s2 times s1 times the identity matrix now our last row operation we represented with the matrix product s3 or multiplying it by the transformation matrix s3 so if you did that if you have s3 s2 s1 a but if you perform the same exact row operations on this guy right here you have s3 s2 s1 times the identity matrix now when you did this when you perform these row operations here this got you to the identity matrix but what are these going to get you to this is when you just perform the same exact row operations you perform on a to get to the identity matrix if you perform those same exact row operations on the identity matrix what you get you get this guy right here right anything times the identity matrix is going to be equal to itself so what got what is that right there that is a inverse that is a inverse so we have a generalized way of figuring out the inverse for the inverse for transformation matrix what I can do is let's say as some transformation matrix a I can set up an Augmented matrix where I put the identity matrix right there just like that and I perform a bunch of row operations I perform a bunch of row operations and you could represent them as matrix products but you perform a bunch of row operations on all of them you perform the same operations you perform on a as you would do on the identity matrix and by the time you have a in as an identity matrix you have a in reduced row echelon form at time a is like that your identity matrix is going to having perform the same exact operations on it it is going to be transformed into in verse and this is a very useful tool - for solving actual inverses now I've explained kind of the theoretical reason why this works in the next video we'll actually solve this maybe we'll do it for this the example that I started off with in this video