 Hi, I'm Zor. Welcome to Unizor Education. Previous lecture was about determinants for 3x3 methods. So, basically, we started with a system of equations like this. Three linear equations with three variables, x1, x2, and x3. So, this is the system of three linear equations with three variables. And we were talking about condition on coefficients a with different indices. The first index is a column and as a row and the second index is a column. So, we were talking about condition on indices which assure us that the system has a unique solution. Or, if you wish, you can actually interpret this system as a transformation of coordinates from x-coordinates to y-coordinates in the 3D, in the three-dimensional space. And, basically, the condition is when this particular transformation can be reversed. For instance, the rotation can be reversed but projection cannot. Okay, so, the condition which we were talking about is basically in the formula which combines all these coefficients into an expression which is called determinant. So, let me just repeat what this particular determinant is. A is a matrix of coefficients. So, the matrix is basically 1-1, 1-2, 1-3, 2-1, 2-2, 2-3, 3-1, 3-3, 3-2, and 3-3. So, this is the matrix we were talking about. And determinant of this matrix is defined as a relatively long expression which I'm going to write right now. But it's very easy to remember it. It has positive and negative members. Positive members are main diagonal, which is a-1-1, a-2-2, a-3-3. And two triangles which have a short side parallel to the main diagonal. This one and this one. So, it's a-1-2, a-2-3, a-3-1. And this one, a-1-3, a-2-1, and a-3-2. And then, 3-negative with alternate diagonal which is a-1-3, a-2-2, a-3-1. And again, triangles which have the short side parallel. So, this is the main diagonal. So, we're talking about this triangle and this triangle. So, it would be minus a-1-1, a-2-3, a-3-2, and minus a-1-2. A-2-1, a-3-3. So, that's the formula. Now, it looks cumbersome and, well, and it is. But it has really wonderful properties. First of all, as I was saying, equality of this particular expression to zero means that the system does not have a unique solution. The system of linear equations or the transformation is not reversible. You cannot find x from y. Basically, it's exactly the same thing as the system not having a unique solution. So, that property by itself is extremely interesting. We saw that there were such properties with 2 by 2 matrix. And this is an expression for 3 by 3 matrix. So, for any square matrix, actually, we have this type of expression, the determinant, which basically determines whether the system with these coefficients does or does not have a unique solution. But there are some other properties and some simple properties I will explain in this particular lecture. So, the property number one, if you will just look at this particular formula, you will see that each element of the matrix participates in this formula. I mean, anyone, like A23. Well, you'll find A23. Now, it also participates exactly twice. Once with a member with a positive sign and once with a member with a negative sign. I was talking about A23, for instance, and any other element. Let's say A12, here is the minus and here is the plus. No more A12 in the formula. So, each element of the, each coefficient in this system participates exactly twice. One with a positive member of the sum and one with a negative. So, that's number one property. Now, what's also interesting is that in each member of this sum, we have, each member contains the product of three elements of the matrix, right? What's interesting is that all these three elements in each member of the sum always belong to different rows and different columns. You see, the rows, for instance, this element. Rows are the first row, the second row, the third row. Columns, the first column, the third column, the second column. So, it's always different rows and different columns. That's why we are mixing together and nicely mixing together basically all these coefficients in this formula. So, the magic doesn't really stop here. There is some other interesting property. For instance, I was already talking about this geometrical kind of memorization technique which you can use if you want to remember this formula. It's really very easy. The main diagonal and a couple of triangles with a short side parallel to the main diagonal are with pluses and the alternate diagonal and a couple of triangles with a short side parallel to it was a negative sign. So, that's another thing. And what else? Now, let's think about first from the perspective of the system of the equations. Now, you understand that if, for instance, let's go back to the equation. So, it's not Y, it's not a transformation of coordinates, but this is a system of equations, B1, B2, B3. Now, you understand that if my row of the coefficients, for instance this one, is completely equal to zero, then if the B1 as well is equal to zero, then I don't really have any equation here because obviously it's an identity zero on the left and if all the coefficients are equal to zero, I have zero on the right regardless of x1, x2, x3, which means that it's as if I don't have the first equation at all and I have only two equations with three variables, which obviously is not a good thing to have. It's most likely has the infinite number of solutions or no solutions at all. In any case, you cannot expect a unique solution from a system of two equations with three variables. Now, the same thing actually happens if any particular column is equal to zero. For instance, this column is equal to zero. Then we don't have x2 at all. So, what do we have now? We have three equations with two variables, x1 and x3. Nothing depends on x2 basically, right? So, either again, system doesn't have any solutions or it has an infinite number of solutions, but in any case, it's not a good idea. It's not a good system which we would like to consider. So, it looks like if any row or any column is equal to zero, then this system is no good. Now, does our determinant really gives us exactly the same answer? Well, let's just consider. For instance, I would like to check if this particular column is equal to zero. What happens with the determinant? 1, 2, 2, 2 and 3, 2 are all equal to zero. What happens with this one? This is zero because it's 2, 2. This is zero because there is a 1, 2. This is zero because there is a 3, 2. This is zero because 2, 2. This is zero because there is an a, 3, 2, which is equal to zero. And this one has 1, 2. So, all elements are equal to zero, so the whole sum is equal to zero. So, that's extremely important quality of the determinant if it's, well, basically it reflects the bad property of the system itself. The system is not good if one column is equal to zero of the coefficients and the determinant is equal to zero. It's really a good signification that the system is not good. Now, what are the properties? Well, for instance, I am increasing one particular row of the coefficients by some factor, k, let's say. What happens if I will increase these three numbers in this matrix by the same factor k? Well, obviously, the whole determinant also will be multiplied by this y because these coefficients are participating in every member here. Like, for instance, what if I will increase this second row, right? So, a22 is increasing and these two are not. So, this will be multiplied by the same factor k. Now, in this case, I have 23 also from the second row. In this case, I have 21 from the second row, etc. All members of this sum have exactly one element of the second row and that's why the whole expression will be increased by the same factor. So, if you increase by some factor a row or a column, the same thing actually you can check it, then the determinant will also be increased. Now, this might not be very interesting property, but here is the really interesting property. Again, let's go back and consider the system of our equations. Now, consider for a moment that one of these equations is actually a combination of two others. Let's say if you will multiply this by m and this by n and add them together you will get exactly the first equation, just as an example. Well, what does it mean? It actually means that we have only two real equations. This equation doesn't bring us anything new if it's a linear combination of these two. So, we have two equations with three variables which is definitely the system which is not uniquely solvable, right? So, again, my question is, does the determinant reflect this particular property? So, what if two rows or actually two columns as well of the matrix are in such a relationship to the third one that we can obtain the third one from the first two by their linear combination? Would the determinant be equal to zero in this case? Because we're always telling that the determinant equals to zero signifies that the system is not good, right? So, this is a perfect example of the system which is not good. And let's check out if determinant really reflects this particular quality of the system. Well, in my example which I'm using in the notes for this lecture, I think I'm putting the second column to be a combination of the first and the third. Just as an example, it doesn't really matter which is which, it's exactly the same thing. So, that's what probably I will try to do. So, what if my matrix is such that the first, the second column is a linear combination of the first and the third? Which means that a12 is equal to m times a11 plus m times a13, a22, the second coefficient, exactly the same mix of the first and the third. And this last element in this column is equal to the same linear combination. So, what if I have this particular condition? My question is, would my determinant be equal to zero in this particular case? Just to make sure the determinant really, the equality of the determinant to zero reflects the property of the system to have a unique solution. Well, obviously, we all expect this to be true, so let's just try to prove it. So, I will just substitute instead of these values in my formula these values and let's see what happens, right? So, which should I substitute? I'm substituting this one, a22, a12 and a32 and a22 and a32 and a12. So, these must be substituted with these values. Well, let's just do it. Again, it's a little bit tedious and I already used this particular example in Winter Olympics when people are skiing for 50 kilometers. That's not fun, neither. That's really tiresome and tedious and really checks your stamina. So, let's check my stamina here. All right, so it's a11. Now, instead of this, I have to substitute it with this, right? So, an a33 is remaining and then I have m, a21 plus m, a23, right? So, instead of 22, 11 and 33 remains, but instead of 22, I substituted this. All right, fine, plus. Now, 23 and 31 remain as this and now I have m. Now, 12 is this. So, it's m, a11 plus m, a13 plus. Here, I have to substitute 32. So, a13 and a21 remain as this and instead of 32, I will change it to this one. Okay, now this one, that's actually minus. We've already switched to minus. So, 13 and 31 and m, so it's 22. It's this one, a21 plus m, a23 minus. This one, I have to substitute 32. So, it's 11, 23, m, a31 plus m, a33, right? And the last one, 21 and 33 remain the same and here I have m, a11 plus m, a13. All right, so that's my formula. Now, is it equal to zero? Well, let's just, you know, analyze it. It should. All right, 11, 33 and 21 was a plus sign. 11, 21 and 33 and m was a minus sign, right? So, these are out. These are canceling out. Now, m, a23, so 11, 23 and 33. 11, 23 and 33 was a plus and this is with a minus. So, this goes out. So, this whole thing is out now. Next, 11, 23, 31. 11, 23, 31 with a minus sign. So, this goes out. Now, this whole member now is out, right? And 13, 31 and 23. So, this one, I think, 13, 13, 31, 31 and 23, 23 and plus and minus. This goes with this one. Now, this completely out. Now, this one, 13, 21 and 31. 13, 21 and 31. This goes out. 13 to 133. 13 to 133, this goes out. So, this is completely out and this is completely out. Now, this was already out. That's it. All cancel out. So, as you see, if we have one particular column of the matrix, which, if it can be expressed as a linear combination of two other columns, then the determinant is equal to zero. And again, from other considerations, if you consider just a system of equations and you have the same thing, you know that this is not good. So, basically, it corresponds. So, again, my point is that the determinant is such a nice thing about matrix that it really determines the whole behavior of the system of equations with this matrix as a system of coefficients or transformation of coordinates. So, these properties are extremely important, especially the last one, linear combination of column with another set of columns. That is actually extremely important. So, whenever you have a system of equations, like three equations with three variables, linear equations, you can always, without basically doing any kind of calculations, seeking solutions, et cetera, et cetera, you can always determine whether the system has a unique solution or not by comparing its determinant, the determinant of the coefficients with zero. So, one formula with all these coefficients combined together can give you the result of this particular dilemma. Now, what's interesting is that these determinants not only answer the question whether the system has a solution or not, they also help you to find this particular solution. But this is a subject of the next lecture. So, thanks very much for today. Don't forget that notes for this lecture are on Unisor.com, and I certainly recommend you to read again to go through these notes and to refresh the material. Now, if you sign in to the website, you can also take exams, which is, I believe, very, very important. But obviously, you can browse through any educational material without registering or anything like this. And, by the way, registration is free. All right, that's it for today. Thanks very much and good luck.