 Hi, I'm Zor. Welcome to a new Zor education. I would like to continue talking about matrix determinants. I did spend some time on two by two matrices and Let me just remind you very briefly if you have a system of two equations with two variables and I'm talking about linear equations, of course Then the matrix of coefficients Which has two rows and two columns It has an expression a11 a22 minus a12 a21 as a very important Characteristic of the matrix Now this characteristic actually is so important that whether it's equal to zero or not is sufficient To determine whether the system has a unique solution So if this is a system of two equations, we are talking about unique solution. If this is a transformation of coordinates From x coordinates to y coordinates then This basically is equivalent to whether the transformation is Inversible, I mean if you can find out x1 and x2 as a unique solution based on y1 and y2 That means we can transform it back Right, so from x to y and then if y is given then we can back transform it back to x So this matrix of transformation now it's transformation of coordinates is Again, it's important and this particular determinant whether it is or it is not equal to zero is a characteristic of both the system of equations or Matrix transformation of the coordinates Basically, I might actually say that if this determinant is not equal to zero then the matrix is well good in certain way Because the unique solution of the system exists or the transformation linear transformation is inversible If the matrix has the determinant equal to zero then the system is not good I mean I might say it's bad Basically, it means that system either doesn't have a solution or there are infinite number of solutions But in any way, there is no unique solution Let's put it this way and the transformation which is defined by this matrix from x coordinates to y coordinates is not Inversible now, what's an inverse what's not inversible what's inversible and not inversible transformation of two coordinates It's on the plane basically well for instance if you want to stretch the plane So every vector is multiplied by two. That's a linear transformation and it's inversible Basically, it's like 1 1 why why 1 is equal to some coefficients time x 1 and why 2 is equal to the same for instance coefficients by x 2 However, if you have a projections for instance like y 1 is equal to x 1 and y 2 is equal to 0 for instance Right. So this is not really a good transformation because lots of different vectors are transformed into the same one If their projection is the same right, so This is not a good transformation, right? So the terminal equals to 0 means it's not a good transformation and it's not a good system of equations now I would like to Basically do exactly the same with a three-dimensional case. So I have three coordinates my system of equations looks like this So my matrix of transformation is 3 by 3 instead of 2 by 2 and the question is Does this matrix of the coefficients a 1 1 a 1 2 etc. Which has three rows and three columns? Does it also have this particular characteristic like a determinant for two dimension for 2 by 2 case? Which would be again a very indicative of whether this transformation of coordinates in the three-dimensional space is is good or bad Whether it's reversible so to speak or whether the system of equations Has a unique solution So either or we are talking about the matrix from these coefficients a was indices and the question is Is it good or bad and how to determine whether it's good or bad? And is there anything like a determinant for two-dimensional case which would help us to determine this particular? Quality of the matrix, but first of all the answer is yes, there is such a characteristic And it's called exactly the same way. It's determinant. So the question is what is the terminal for a matrix? Which has three rows and and and three columns so what I'm going to do is I'm going to solve this particular system of three equations and In the process of solving I will basically well determine if I can solve it and it will produce Unique solution then well the system is good and if there is some condition when I cannot do this Then that would be a condition Which I'm looking for and that would be a formula for a determinant What's the condition on all the coefficients the formula basically which combines them together? In an expression which either equal to zero or not equal to zero So if it's equal to zero same as a two-way two case I will say that the system is not good and if it's not equal to zero the system does have a unique solution or transformation of three-dimensional Space is reversible. So how can they solve it? Well, first of all, let me just assume that this is a true system of Three Equations with three coordinates Now for this I need at least one of these coefficients not to be equal to zero because if they're all equal to zero Then it's definitely a bad system because it has a three Equations with only two Variables and it's either doesn't have any solutions or it has zero solutions basically or or it has in some infinite number of solutions or The equation one of the equations is basically a linear combination of two other equations Which means we still have a two-dimensional case. We have two variables independent variables So I assume that one of these For x3 is not equal to zero and just for Definitiveness I assume that a one three not equal to zero. All right at least one of them should be not equal to zero Now, how can I solve it? Well, here is my plan First I will reduce this system of three equations to the system of two equations with two variables I will eliminate x3. How can I do it? Well, if I multiply the first Equation by a two three and the second equation by a one three and subtract Then my x3 will cancel right and then I will do the same with with the first and the third one I will multiply by a three three and I will multiply by a one three and again subtract and Again, I will have another equation in both cases x3 will cancel out and I will have two equations with two variables x1 and x2 All right. So first of all, let's multiply this one by a two three and this one by a One three All right and subtract From the first I will subtract the second. So what will I have? Well, I will have on the left. I will have b1 times a two three minus b two times a one three Well, my most important problem right now is not to make any mistakes because there is a lot of Calculations here which are which are kind of tedious but I would like actually to mention that tedious and long Calculations, they also have their purpose Think about winter Olympics, for instance, you have skiers Cross-country skiers who are going for like 50 kilometers. We think it's fun. I Guarantee it's not fun, but it kind of develops your stamina, right? So all these long-distance calculations develop the stamina for your brain. So brain also has a stamina You have to really be, you know, persevering All right. So the left part I have just done now the right part will be now the X1 coefficient will be a11 times a to 3 minus a to 1 times a13, right? so a11 a23 minus a21 a13 That would be x1 and x2 will be a12 a23 this times this minus this times this minus a22 times a13 That would be x2 and x3 will cancel out Because it will be a13 a23 times a23 minus a23 times a13 so it's zero, right? Okay, fine. This is done Now let's do another thing We will multiply this by a a13 and we will multiply this By a33 Now the first and the third and then we will subtract them My x3 will again cancel out now. What would be on the left on the left? I will have b1 a 33 minus b 3 times a13 equals 2 a11 times a33 minus a31 a13 that would be x1 and coefficient at x2 would be a12 a33 this times this minus this times this minus a32 a13 x2 all right, so now I have reduced my three equations with three variables to two equations with two variables, right? so Now I can solve it, but at the same time I can actually use the previous results. You remember we were actually Researched the system of two equations with two variables and we found that the determinant of The matrix of the coefficients is basically a decisive factor if it's equal to zero Then we have a problem. The system does not have a unique solution So if my determinant of this system has it is equal to zero Basically, I will have some formula for all the coefficients of my initial matrix And if it's equal to zero that means basically that my The system does not have a unique solution, right? So I will just use this so my determinant for this system is This time this minus this times this right? That's the terminal the main diagonal minus alternate diagonal Well, and again now my most important problem is not to make mistakes because I have to Make a lot of multiplication, etc. So just bear with me. I hope I will not make a mistake But any case so this times this that would be a four different members This times this first so I will try to order them in a sequence of the first index So it's a 1 1 a 2 3 a 1 2 and a 3 3. So I will use a 1 1 a 1 2 and then a 2 3 and a 3 3 That's my first number now this times this Would be with a minus sign a 1 1 then a 1 3 a 2 3 and a 3 2 You see I'm ordering these in the order of the first index and if the first index is the same then ordering in the second All right next I multiply this by this and that's a minus sign Now first I will have 1 2 1 3 2 1 and 3 3 and Finally the fourth member is with a plus sign because this is minus and this is minus. So it's a plus This and this I have two actually 1 3 Then 2 1 and 3 2 All right, so this is this multiplied by this now I have to subtract this multiplied by this so it's minus sign Okay, first by first. So it's 1 1 1 2 2 3 and 3 3 right 1 1 2 2 3 and 3 3 Okay, that was with a minus sign now this times this it's minus, but then since I have minus in front it would be plus The smallest one 1 2 index then 1 3 Then 2 3 and 3 1 Now this member times this Now this is the minus, but there is a minus in front. So it plus 1 1 would be the smallest index Then 1 3 Then 2 2 and Then 3 3 Finally this time this it's a plus, but there is a minus in front of it. So it would be in the game there is a double 1 3 and then 2 2 and 3 1 I Think I did not make a mistake. I hope at least So what can we do about this? Well? this is an Expression of the determinant of this particular matrix, right? So if it's equal to zero, I'm in troubles in the sense that the whole system does not have a good solution, right? so I actually can just analyze this and before Equaging it to zero before actually researching this stuff I would like to simplify it a little bit and here's how well first of all I see that there is something which is supposed to cancel out The first one and and and this one you see they are the same cancel out Plus and minus all right now What's interesting is that all other members contain a 1 3 as a factor you see a 1 3 a 1 3 a 1 3 a 1 3 a 1 3 and a 1 3, okay, so I have Basically assumed that a 1 3 is not equal to 0 because I have to assume that's one of these coefficients is not equal to 0 Right, so if I am asking whether something is equal to 0 or not And I can factor out a 1 3 which is not equal to 0 So whatever will be left after I factor it out is supposed to be either Equal to 0 or not because a 1 3 is not equal to 0 which means I can basically consider instead of this particular expression the one which is This divided factored out one a 1 3 right so without a 1 3 and I will also do it Just regrouping first I will get all pluses and then I will get all minuses right so all pluses are And I will try to arrange it again in sequence this plus this plus and this plus This is the smallest a 1 1, so I will start with a 1 1 Now a 1 3 would be Factor out here here here Here and and here all right, so I will do it without a 1 3 now because it's not equal to 0 so a 1 1 a 2 2 a 3 3 that's one Then I will have this one was a plus sign Because it has the first one a 1 2 and then trying again to be in sequence. Oh No, there is no There is no a 1 3 anyone Yes a 2 3 and a 3 1 So this is done. This is done and this is another with a plus A 1 3 a 2 1 a 3 2, okay all pluses are done now all minuses minus So this is the smallest index it's minus a 1 1 a 2 3 a 3 2 Now this is minus a 1 2 a 2 1 a 3 3 and This is minus a 1 3 a 2 2 a 3 1 all right, so let me wipe out everything else and basically I do have the expression which I can consider as a very important characteristic of the matrix of the coefficients if this expression is equal to 0 Then I can say that my system does not have a solution does not have a good solution unique solution all right now and this is actually by definition the The entity which is called the terminate of this matrix So I know that if this is equal to 0 my matrix does not have my transformation is not Unique transformation my system of equations does not have a unique solution. It's okay, right? So let me just talk about this particular expression. It looks a little bit Well cumbersome, I don't know Complicated However, I would like actually to show you the geometry of this thing that would make Make it easier to understand actually how it's all done. Now, let's just Put it in perspective This is a matrix of coefficients The interesting interesting thing is I actually do remember by heart this formula I'll just show you how I remember it now look at the three positive members now this one is Main diagonal right a while one eight or two and a three three so we start from the main diagonal now a one two which is this a Two three which is this and a three one is this this is a triangle Which has a short side parallel to the main diagonal now this one one three Which is this? To one which is this and three two which is this? This is another triangle with a short side parallel all right, so All positive members are the main diagonal and two triangles With the short side parallel to the main diagonal now, let's talk about the negative sides Okay We start with this now. It's a one one a two three which is this one and A three two which is this one right? Okay, the red one so it's this triangle one two two one and Three three this is this triangle and finally one three Two two and this is the alternate diagonal now all negative members are Alternate diagonal and again two triangles with a short side parallel to it So if you just draw this particular matrix as a square basically so two diagonals main diagonal with the plus Alternate diagonal with the minus and then everything which is parallel all triangles, which are we just with the side parallel to main Diagonal are with the positive sign all triangles with All triangles like this one the red one which are which are the side parallel to this diagonal are short ones With a negative sign So that's how I remembered now this formula actually is the definition of the determinant for this particular matrix and next lecture I will probably spend to analyze the formula its properties and How to how it affects the system of coordinates transformation and and Equations systems of linear equations All right, so now the same thing actually which I was just explaining is in notes on unisor.com And I do recommend you to go through this again just by yourself It might actually contain certain more details, but basically it's exactly the same thing And again next lecture will be dedicated to properties of this determinant for three by three matrices and Some comments about it. That's it for today. Thank you very much and good luck