 So, now that we have got this clear for a general matrix it is rectangular it is you know this particular result here we talked about a fat matrix and we have shown you that you will always have a norm 0 solution. Let us now turn our attention towards square matrices and see what we can say. So, now let us say Ax is equal to 0 is still the equation in focus, but now this A is a square matrix all right. We will get back to the rectangular case once we have covered this because there are a couple of interesting and important observations to be made for the case when it is square. So, we are going to make this first claim about this system ok. So, when I use a symbol like this it means it is an if and only if condition it is a both sided implication ok. It means it is necessary and sufficient quite counter intuitively it might appear although while saying we say necessary and sufficient if and only if, but if actually corresponds to sufficiency only if corresponds to necessity yeah. Just think about the semantics the language a bit and will be clear right. So, when something is sufficient you say if this is true then this happens it will definitely happen if this is true, but it is if only if this is true then this happens then it is a necessary condition. It means if it does not happen when you are saying sufficiency it means that you are probably being a little more conservative. You are making more leave a more room for things and saying ok at least if this much is there then it is guaranteed to happen and in the process you might be actually laying out more requirements than are necessary, but when you are saying something is necessary it means if you take even the slightest amount of those constraints away then that fact would not happen that fact would not follow right. So, that is necessity the only if part the sufficiency the if part when I write symbols like this it generally implies if and only if sufficient and necessary implies both ways. It is the same symbol right linguistically or symbolically. So, what I am going to now make a claim for is the following. So, let us say we have R R E F of A is equal to the identity it is a square matrix. So, R R E F of identity R R E F being identity means it has no 0 rows every row has a leading one. This is tantamount to saying it turns out that the only solution to A x is equal to 0 is x is equal to 0 a very important result. So, how do we go about proving this? We have to prove both sided right we cannot just prove one side of it because it is a both sided implication. So, how do we start we have to assume one of the sides is true and then prove the other and reverse the process and do the other same with the other right. So, let us say so, I will give you a sketch of the proof suppose R R E F of A is equal to identity what does this mean that there must be a series of operations that takes you to the identity. So, there exists M such that M inverse exists and M A is equal to identity right. So, consider A x is equal to 0 and M A x is equal to M 0 which is nothing but 0 you agree I hope that this and this are equivalent systems of course, follows from the definition it is equivalent is not it because that is how you get to equivalent systems through non-singular P multiplications right these are tantamount to row operations. In fact, this is not just any arbitrary M this is a series of elementary row operations which have been brought up together to get this M, but we do not even need that now. What we can say is therefore, let me erase this part and continue the proof here. So, the proof continued here what can we then say that A x is equal to 0 what is M A x its identity times x and identity x is equal to 0 are therefore, equivalent systems. So, what is the solution set of the second equation. So, solution set of I x is equal to 0 is S given by x such that I x is equal to 0 which is nothing but x such that x is equal to 0. So, that is the only solution right now if this set is the same as a solution set of this this is also equal to solution set of A x is equal to 0. So, therefore, A x is equal to 0 also has only a single solution which is x is equal to 0 the trivial solution and nothing more than that because that is our definition that is the way we have described it when you cook up equivalent systems they share the same solution sets like we approved in the previous lecture right. So, one side of the proof is done good we will now try to complete the other side in this area. So, now we have to assume the opposite we have to show that the RREF is identity when in fact A x is equal to 0 has only one solution ok. So, suppose A x is equal to 0 has only x is equal to 0 as its solution right. Now what do we have to show we have to show that RREF of A is equal to identity what we shall do is we shall assume the contrary. So, these are proof techniques that you should familiarize yourselves with. So, suppose that the RREF of A is anything but the identity yeah. So, let us suppose not which means that suppose despite the fact that A x is equal to 0 has only x is equal to 0 as its solution the RREF of A turns out to be something other than the identity we will have to contradict this we will have to prove that this is an absurd proposition it cannot be true. So, let us try and oppose this. So, suppose RREF of A is not equal to identity that is RREF of A is not equal to identity what then could it possibly be see it can have at most n pivot variables. If it has n pivot variables by the pigeonhole principle because k 1 is strictly less than k 2 is strictly less than k 3 it must be only the identity. The only way that this is not the identity is if you have fewer than n what pivot elements. So, that means there exist are less than n pivot variables or you can say leading ones. What is the total number of variables then n the number of pivot variables strictly less than n. Therefore, there exist free variables, but that is absurd right why because we have already assumed that A x is equal to 0 can have only the 0 solution. Now the moment you allow the moment you allow free variables you are basically allowing non-zero solutions to this equation yeah because this means that x not equal to 0 ok let us say there exist x not equal to 0 such that A x is equal to 0 which is a contradiction. So, where does this contradiction arise from from the fact that we have assumed it is not the case. So, the only possibility is that the row reduced echelon form of A has to be nothing but the identity right. So, therefore, it is an if and only if condition this claim that we have talked about here is an if and only if condition right any doubts please ask ok. So, next we are still going to dwell on this A x is equal to 0 with a being square and we are going to look at another equivalence I hope I can erase this part ok. We are going to have another equivalent condition which is R R E F of A is equal to identity and it means that there exists A bar such that A bar A is equal to A A bar is equal to identity which means that is another way of saying that A has an inverse because this is the definition of the existence of an inverse. If there exists an A bar such that A bar is equal to A A bar is equal to identity it is an inverse right. So, that means A is invertible the point that I am making is if the R R E F of a matrix is the identity then it is invertible again it is an if and only if condition. So, it is only fair that we prove both sides of this assertion ok. So, suppose we start with let us say we start with any of those two sides suppose we start with R R E F. So, suppose R R E F of A is equal to identity ok I might not have erased it turns out because the first few steps are going to be exactly the same this implies there exists M such that M inverse exists and M A is equal to. So, I am not writing the sizes of this matrices I hope you understand that M is also a square matrix of size n cross n yeah such that this is true alright. Now, already I have one part of it what I mean by that is M A is identity if I can show that A M is also identity then M would be exactly the candidate for the inverse is it not right. So, look at A M next. So, based on what we know we go into the unknown ok. So, this is where we start. So, M A is equal to identity implies I am going to just hit it with an M on the right and say M A M because of the associativity of the multiplication of square matrices yeah any matrix for that matter I can actually combine them in any way I like and this is going to be I times M is equal to M let us continue that here I am going to now continue it here. So, what do we have now instead of taking this M A I am going to take this A M and I am going to write M times A M is equal to M. Now, of course, this fellow M definitely has an inverse I may not know about whether A has an inverse or not, but M definitely has an inverse by dint of this step that I have assumed here I mean I am not assumed it this is true if it has to have an RREF form then there must be a non invertible matrix which hits this A and takes it to the identity. So, what can I say let us hit it with I am not writing all the steps you can fill in the blanks in between I am just hitting it with M inverse on the left on both sides. So, I am just hitting it with M inverse M A M is equal to M inverse M right and what turns out immediately I have the implication these are pulverized into identity. So, I have A M is equal to identity. So, I already have M A is equal to identity now I have A M is equal to identity all that I need to show is that the inverse of a matrix is unique and then M is the inverse up until this point I can maybe say M is an inverse of A if I want to claim it is the inverse of A I need to show that it is unique I mean for any matrix if you have both left and right inverses and the matrices inverse must be unique I will leave that to you as an exercise to show try that it is not very difficult it is something similar to what we have done in the previous lecture when we showed this left and right inverse very similar. So, take that as an exercise show that the inverse of a square matrix is unique if you do show that then you will know based on this exercise that M is the inverse of A. So, I have actually shown more than what this claims this claims says that this must be invertible I have also given you a constructive method of actually explicitly saying what that inverse will be it is exactly that matrix which takes it to the identity in the RRA So, the RRAF is identity it is invertible, but now I need to show the other way as well. So, now my starting point is that there exists suppose there exists A bar such that A bar A is equal to A A bar is equal to identity. How do you think we will go about this what is the step? So, what is the first step that we do how do we address this how do we prove that the RRAF because the inversion process yeah we do not yet know what to do remember you cannot use this fact now that will be like the chicken and egg type of thing right circular logic. So, you have to know that how to massage this to an equivalent condition for the RRAF being identity we have already seen one such condition just a while back if I can show that this immediately leads to the implication that A x is equal to 0 has only the trivial solution is that not also an equivalent condition for the RRAF of A being identity. So, I will address it in an indirect fashion instead of showing that whenever you have an inverse for A the RRAF will be identity I will show instead an equivalence condition of the RRAF being identity which is just what I proved a while back. So, now let us reset my goal to the following that if this is given I am required to show that the solution of A x is equal to 0 must be x is equal to 0 how do I do that? So, what is your suggestion how do I go about this? So, let us start with A x is equal to 0 and then so consider A x is equal to 0 then what happens? So, because this inverse exists I can hit it with an inverse here right and that is a non-singular matrix. So, it does not change does not alter the condition yeah. So, I can just say A bar A x is equal to A bar 0 which is nothing but 0 that is an implication let us continue it here and therefore, we have I x is equal to 0 which implies that x cannot help but B 0, but this just a while back we have shown is equivalent to saying that RRAF of A is the identity so therefore, it is it works both ways. In summary in view of all that we have discussed so far we can say that the following conditions are equivalent what are they? RREF of A so let us say for A which is a square matrix n cross n RREF of A is equal to identity then A x is equal to 0 has a unique solution given by x is equal to 0 and A is invertible yeah. So, that is the big result probably I should box it that is what we have seen right a very important result in linear algebra for square systems something that you are already familiar with I am sure but you may not have looked at in this fashion you might have looked at those determinants and the explicit formulae for inverses this does not talk about how to find the inverse well explicitly at least in terms of some formula like using determinants and adjugates and stuff yeah, but it still says the same thing yeah any of these checks would suffice ok this is clear any doubts so far all right.