 Let us say you have this original matrix A on this you have used several of these elementary row operations let us say and let us say this is E A hat or A bar I have been using the bar notation confusing A bar right. Now this entire thing can be thought of as a common M. So, if each of these individual chunks are invertible is not the whole M invertible. In fact, we know that if E I are invertible then so is M right. In fact, we can also say that M inverse is exactly equal to what it gets reversed again you can view it as an action. What does it mean to have acted like this? The first to act on A was E 1 then the subsequently it was acted upon by E 2 then E 3 by E 4 and so on. So, if you want to undo you should start with the last misdeed that you have done. If you want to undo your past actions and want to sort of be penitent about it start undoing from your most recent crime the effects of that right. So, that is what you do. So, you first hit it with the last action that you have done right. So, what is this equal to then right. So, this is again a point that is beyond any debate or anything there is no debate about this. So, we are happy with this because now we exactly have a way to go from A to A bar through a matrix M and a way to go from A bar to A through the matrix M inverse which is also another M I mean both are invertible right. So, it turns out that this elementary row operations provide us with a root to get from one system to another equivalent system. But the question still remains what are these so called nice forms that we are shooting for what would it what would make us happy right philosophical question, but we will not address the philosophy in this class like in general what would make us happy no one knows. But yeah in this class at least we know what would make us happy in so far as this form of this equation is concerned. So, we have A x is equal to b any arbitrary structure. Let us give a particular form or shape and say that this is desirable we will see when we actually go about solving it we will see why it is a desirable form. So, we give a particular form a name in fact there is no reason to retain this row reduced form suppose we have been given a matrix A first we have to understand what is this row reduced form and then we go to try and see if there is a way to get from any given A B pair to the row reduced form of this A ok. So, what is the definition of this row reduced form matrix say R which is M cross N is said to be in the row reduced form if the following hold first the first nonzero entry in each row is 1 we will call this the leading one ok. If the first nonzero entry as you go on reading where I am saying reading I understand it from left to right ok. So, I am not referring to languages like Hebrew or Arabic where you read from right to left reading from left to right and we are looking for the first nonzero entry. So, the first nonzero entry has to be 1 yeah. So, if any matrix checks out against the first criteria then we go out checking for the second criteria and verifying whether that is also met if that is met then it is row reduced what is the second criteria other entries in a column containing leading 1 must be 0 ok. If that is a bit too much to parse let me give you an example and maybe along the way I will also cook up some non examples. So, let us say you have a matrix like so 0 0 1 5 7 and 0 1 0 4 2 what should I do 1 ok 1 or let us make this whole zeros yeah no problem 1 0 5 7 6 this is not a row reduced matrix the way I can I mean what would I have had to do or what would I have had to change in this matrix to make it row reduced. See if I look at it the first nonzero entry in each row is 1. So, it checks out against the first condition. However, it is the second condition that is important to us now because we have checked the first condition already. So, what are we going to look for now? Now that we figured out that every nonzero row has a leading 1. Now we are going to focus on the columns containing those leading ones. So, what are the columns containing the leading ones? This is a column containing a leading one this is a column containing a leading one and this is a column containing a leading one. So, now check against the second condition you see the first one every other entry in that column is 0 we are good. Second one also third one this falls the party. So, if I had this as 0 this is a row reduced matrix. So, you have seen in one shot a non-example first and now you see an example of a row reduced matrix any matrix that meets these two conditions is a row reduced matrix. So, this is again a definition whenever I give you a definition you should brace yourself up because some result is coming up that is the reason we are setting it up like that we have defined something as a row reduced matrix. So, what is the result that is coming up? Let us call it another proposition every system of equations a x is equal to b can be transformed to x is equal to b hat where r is in row let us just use a short form r r f r is in row reduced form by what means through elementary row operations on a b pair. What I am saying is of course goes without saying let us say a is m cross n so is r because we are hitting it with only square matrices right. So, I am saying that you give me any system of equations I am going to be always able to convert it in such a way that this r matrix will look something like this probably and you might wonder why it is a nicer form again I am not sure we will have time for that in today's lecture maybe we will see it in the next lecture when we actually go ahead and solve one equation one system of equations why this is a much more desirable form than any arbitrary form of the a matrix that you can have right. But the point is that you will always be able to get from any given a matrix to a row reduced form by just using those three sets of operations that we have described or maybe even two in this case as we shall see. In fact, we will see that just the first two suffice I will not give you again a very formal or a complete proof, but I will just give you kind of a constructive insight into what goes into this operation. So, why is this true what is the idea behind this that takes us from any arbitrary a to a row reduced form first how do you think you are going to get or one as the leading non-zero entry in every row given that you have a matrix a like so let us say some 0 then some star here that is the first non-zero entry here in this row let us say in this row you have 0 0 that is the first non-zero entry here let us say here you have this as the first sorry first non-zero entry here some such structure right some such matrix first idea would be to turn this fellow to a unity what elementary row operation would allow me to do that scaling by by the reciprocal of that first non-zero entry right. So, what we need essentially in this number system of ours is the ability to do divisions which as of now might appear very trivial, but as we shall see maybe in four five lectures from now that we will not always be dealing with systems of equations that are in real numbers, but over more abstract number systems in which case the division is not always obvious. For example, if I ask you to solve for integers you cannot divide two integers and always get another integer right. So, you cannot use this sort of technique to get integral solutions to what we call the Dauphan-Tien equations. So, for now we will assume that we are allowed to scale and still you know real numbers of course unless you are taking 0 you can divide by anything and this is obviously the non-zero entry. So, you can obviously scale it by its reciprocal. So, what do you do basically here is two ways we can go about it we can say ok. First let me look at all non-zero rows look at their first non-zero entries and scale them by their reciprocals do we do that? Do we just hit them with operations of the first kind throughout and then try to meet the second condition like 0 out the entries below that leading one. How do we do this? What is the smart way of doing it? So, it should not be like you use the first kind of operation which is scaling on all the non-zero rows first because you might be wasting your computations right because you might have just scale this and maybe on another row also this particular column happens to have the leading one and because you have made it a leading one you have wasted that because anyway this is going to become a 0. So, later on the leading one corresponding to this row will not be this. So, if you have just gone about scaling all the first non-zero entries above in the rows and acting on just the first scaling type of matrices you might have actually wasted your efforts. So, the way to go about it is first this is the constructive way to do it. So, first you have this A matrix first hit it with one operation of the first kind on the first row then you have to 0 out potentially m minus 1 entries. So, that is basically addition of something times the first row plus that row what something what is that something that something is exactly the one times what will take you to this. So, if this entry is 5 then you take minus 5 times the first row plus this makes sense right. So, you potentially have one operation of the first kind that is E 1 and then you take potentially m minus 1 operations of E 2 right E 2 m minus 1 sorry m minus 1 operations. So, this would hopefully allow you to get at least one particular leading one and every entry below that leading one now gets to 0. So, this only takes care of the first. Now, you do that for the second row. So, you keep repeating this. So, you have m operations and you keep repeating this for as many non-zero rows as are there. Assuming that you do not have trivial equations and after all these operations they have survived you might have to do this m number of times once for each row or if some of them have already nullified themselves then your efforts will be a little reduced, but this already you see it has the makings of an algorithm you can code it you do not need any observation or any physical understanding of what is going on or you know guessing what numbers what is the smart choice of numbers and all this can be coded by a computer right. So, you have m this m operations take care of the first and you might have to do m squared such matrices, but eventually you will get there and each of these individually invertible, but what will happen after you have gotten to this point is this really the final form that we are looking for is this where we stop right it turns out we can do one better because you see we have not used the third tool that we have developed or spoken about which is the exchanging. So, what we have just seen is row reduced form now we are going to see a refinement to that form. So, at least this proposition I hope is clear given any system of equations a x is equal to b at least real numbers you can always get to the row reduced form right this is this gives us some hope this gives us some promise that things can be simplified and made to look nice, but we will go one better now we will say row reduced echelon form and in fact in MATLAB there is a command RREF you can just try it right. So, this is RREF right row reduced echelon form. So, again it warrants a definition. So, what is the definition of a row reduced echelon form a matrix right matrix R what should I call it R tilde is said to be in row reduced echelon form do you know what echelon means the upper echelons of the society you know some sort of a hierarchy that you construct layers and so on and so forth. So, echelons you know often using literary sense the upper echelons he is among the highest echelons of writers or artists or whatever. So, row reduced echelon form clearly then we have to get some spatial hierarchy in this matrix that should sort of ring a bell. So, what is it a matrix R tilde is said to be in the row reduced echelon form if first it is R tilde. So, I am not going to redefine that because I already know what is a row reduced matrix. So, it already meets the condition for a row reduced matrix, but so every row reduced echelon form matrix is it this no yeah is it this I mean is it row reduced how does this work is every row reduced matrix row reduced echelon form or is every row reduced echelon form row reduced. So, you have to understand what includes what right what the more the conditions you impose this is going to be fundamental to our understanding. See later on we will understand structures such as rings integral domains fields and so on and so forth and we will see that you will add on more properties it is like an admission test or some qualification criteria or some selection right. So, first you clear your 10th standard, but there are lots of people who clear the 10th standard, but never go on to complete their 12th. Then you complete the 12th and then you complete the graduation then post graduation whatever whatever is whatever you like right. So, if you are selecting certain criteria saying I only want people who have cleared a graduation level then already those who have cleared just the 10th standard are ruled out by that qualification criteria right. So, they are not qualified qualification is nothing about your knowledge qualification is about selection some selection criteria that you put in place. So, when you are imposing this it means this must be true right. So, it must be contained within the set of row reduced matrices already, but apart from that it must also have some specific features. So, not every row reduced matrix will qualify to be called as a matrix in the row reduced echelon form. What is the additional criteria or what are the additional criteria? First every row of all zeros must be below non-zero rows. You can already see where it fails that test this example there is a row of all zeros. So, by rights it should have been below every other non-zero row, but there is a non-zero row sitting below this which prevents this from being row reduced echelon form ok. So, let us flip them 1 0 0 7 6 I will remember that 1 0 0 7 6 and then you have 0 0 0 0 0 and so far based on what we have defined here it seems we are doing ok, but hang on there is one more criteria to go which might look a little more involved, but it really is not that difficult to grasp. What is it? Is this a part of the board visible? It is I think I can see, but is it visible to everyone at the back? Good. Suppose, k 1 k 2 k r are the positions of the leading ones in the r non-zero rows. So, in this case the total number of rows is 4, but the number of non-zero rows is 3. So, r will be 3 all right. So, in this case r is clearly equal to 3 from that notation here just trying to connect that. So, that you do not feel like we are just using too many notations ok. Suppose, k 1 k 2 k 3 till k r are the positions of the leading ones in the r non-zero rows what do I mean by position? If you are reading from the left this position is 3 k 1 is 3 k 2 is 2 k 3 is 1 all right. What we need here is then k 1 is less than strictly less than k 2 less than k 3 and k r that is the echelon ok along with the this one of course. So, the echelon means the zeros have already been brought to the bottom of the heap. Yeah, you do not count your 0 you have no value anyway. So, bottom of the heap and then the leading ones what happens? This is not in the row reduced echelon form because the leading one here k 1 is 3 k 2 is 2 and k 3 is 1 which is at exactly opposite of this. In fact, it might have been any jumbled order also it still would not have been like so. So, if I want to do this what do I have to do? The third kind of operation there is right. So, what is the big picture what is the take home message from all this the last point we will make in today's lecture? What is the result that we had gotten after the definition of the row reduced form? That every matrix A can be massaged through these elementary row operations into a row reduced form. Now, I will go one step ahead and say that every matrix A no matter in what form you give it to me I shall be able to massage it to not just the row reduced form, but the row reduced echelon form through just those elementary row operations and nothing more or less. So, it turns out that these elementary row operations are the perfect construction that I need at my disposal to get to a so called nice looking form. And just to give you a clear idea of what a nice looking form for this would be it would be 1 0 0 7 6 0 1 0 4 2 0 0 1 5 7 and 0 0 0 0 0. But to get from here if you have already seen and you already agree with me that any matrix can be gotten to this form then to get from here to here is just a swapping just exchanging rows first you exchange this first with this and then you exchange this I mean maybe this with this yeah here you do not need to touch the second one you can just swap the first and the third right and you can get to the row reduced echelon form. So, therefore the final result that I am going to put on the board and then bring it to an end here today is this every matrix A which is m cross n can be reduced to R R E F through elementary I could have just written E R O elementary row operations. In fact this also has a name I could have shortened it a bit yeah when two matrices are related by elementary row operations we call them as row equivalent yeah. So, just remember that term so you have seen equivalence of systems of equations and this row equivalent is the equivalence of matrices correspondingly corresponding to those equations right. So, you have A X is equal to B you can get from A to R or you can get to get from A to R till day or R yeah this is what we call R till day this is let us say R yeah. So, every matrix you can do this with is the next lecture we shall see what is the benefit of getting to this row reduced echelon form why does this help us in quickly solving or readily seeing what the solution of R X or R till day X is equal to B hat or B till day is going to look like we can just classify them and just sort of characterize all possible solutions of this system of equations and also say when the solution will not exist straight away and it does not matter whether it is a square system or rectangular system we can always talk about it in the general sense when n is not equal to m as well. Thank you.