 Hi, I'm Zor. Welcome to a new Zor education. We continue talking about matrices, and here is one very important matrix characteristic, which I would like to talk about, to explain basically what it is. It's called determinant out. Well, obviously, it's related to the word determinants. It determines the quality of the matrix if you wish. Another important thing is mostly, we will be talking about square matrix matrices, which has the same number of rows and columns. Mostly, we will be talking about either two-by-two matrices or three-by-three matrices. Obviously, we can talk about other things as well, but these are our major focus of attention. This particular lecture is about determinant of two-by-two matrices. I usually try to put some philosophical foundation for everything I'm trying to introduce. Here is my philosophical introduction to the concept of the determinant of the matrix. With numbers, we have something which is called absolute value. For instance, absolute value of 5 is 5, absolute value of minus 5 is also 5. Now, what does it actually signify? Well, if this particular value, absolute value of the number is equal to 0, then it's not exactly the same in its rights, basically number as any other. So 0 is not the same number as everything else. It does have its own peculiarity. For instance, no matter what number is given, if you multiply it by 0, it will give you 0. There is no other number which would give exactly the same result. In any other case, the result would depend on both, on this number and on this number, but with 0 it's only on this one. So there is some special thing about 0. And another thing, if you add 0 to any number, it will retain basically the value. Now, so what I would like to say is that absolute value of the number has this very interesting characteristic. Is it like the number like any other number? Or it has its peculiarities, some characteristic properties of the number. And those which have this equal to 0 are treated differently than the rest. Now, with vectors, we also have something similar to this. With vectors, we have the length, right? Now, the length, so same thing. If length is equal to 0, then adding to any other vector would not change the value of that other vector, right? And if you multiply it, if you scale this vector by any factor, it will be the same vector. Scaling of the new vector will give you a new vector regardless of the scale, which is not the same as anything else. Any other vector would change with the scaling, right? But no vector will not. So again, this is a specific property of the vector which kind of differentiates this from anything else. Well, similar property exists among square matrices. And it's called the terminal. Now, this is philosophy. Now, let's go to algebra. And we are talking about matrices of the 2 by 2 sides. So let's talk about an equation, system of two equations. You already seen it when I introduced the matrices. So two variables, x1 and x2, are unknowns. All other coefficients are known. And we were talking about that the matrix of coefficients determines, basically, the transformation from unknowns to these two constants b. Now, well, let's solve it. As we usually solve the system of two equations with two variables. Now, how can we solve it? Well, very easily. Let's say we want to solve it for x1. So we have to get rid of x2. So let's multiply this by a22. And this will multiply by a12 and subtract. Then this would be a12 times a22 with x2. And this will do the same thing. So when I subtract, it will reduce. So what I will have is b1, a12 minus b2, a22, I'm sorry, minus b2, a12 equals a11 times a22 minus a21, a12, x1. From which we can derive x1 is equal to b1, a22, minus b2, a12 divided by a11, a22, minus a21, a12. Great. Now, let us find the x2. Now for this, I will multiply this by a21, this by a11, and subtract. What will I have? I will have b1 times a21 minus b2 times a11 equals a11, a21. No, this will reduce. So I have only x2. So it's a12, a21 minus a22, a11, x2. So x2 is equal to this divided by this. But I would like to change the sign. So my denominator would be exactly the same as this one. So it would be a11, a22, minus a12, a21. And numerator I will also change the sign. So it would be b2, a11, minus b1, a21. OK. Now, in both cases, as you see, the denominator is exactly the same. Now, this is a21 times a12. This is another way around. But these are all real numbers. So the commutative rule is working. So we have exactly the same thing in the denominator. Now, what does it signify, basically? Well, first of all, it signifies that for a system of two linear equations with two variables, this number, which is just a function of the coefficients, it's this times this, minus this times this, is very important. If it's not equal to 0, then we can actually solve the whole equation. You see, I have divided by this expression for x1 and for x2. I cannot really do it if it's 0, right? So equality of 0, this particular expression makes the whole system, well, different than other systems, than other linear systems. I cannot have a proper solution to this system. In some cases, I might have an infinite number of solutions. In some cases, I can have no solutions at all. We will investigate it in some other way. But what's important is that this characteristic of the equation, this function of the coefficients, is extremely important. And it actually determines whether the system is solvable in a normal way or it's not. Now, let's not forget that this matrix A1, A2, A2, A2, A2, determines the transformation from x to b, right? So the transformation is not really the good transformation if this expression is equal to 0. So this determines the quality of the transformation itself. Well, yes, it determines the quality of the matrix. And that's why it's called the determinant. So this is determinant by definition of this matrix. And here is how we will write it. Determinant of matrix A11, A12, A21, A22 equals to 1, 1, 2, 2, minus 1, 2, 2, 1. So by definition, this expression for a 2 by 2 matrix is called determinant. And one of the properties of this determinant is that if it's equal to 0, then this system cannot be really solved easily. Well, not easily. There is no single good solution to the system. In some cases, as I was saying, it might be infinite number of solutions. In some cases, it can be no solutions at all. So this plays extremely important role for matrices. For 2 by 2 matrices, this is an expression. There is also expression for 3 by 3 and for all other matrices. But right now, today, it's only for 2 by 2. Maybe some other day, we will do some other dimensions. So it plays exactly the same role for matrices, for square matrices. And in this case, it's 2 by 2 square matrices. As the length of the vector plays for the vector, or absolute value plays for the numbers. And now, after I have introduced this particular characteristic, a numerical characteristic of a 2 by 2 square matrix, well, let's investigate the properties of this. That's very important. So let's forget about this. So this is a definition. And now we are talking about properties. OK. The property number 1, if there is at least one row or one column in this matrix, which is equal to 0, then determinant is equal to 0. So matrices of this type or this type. So when one row, first or second row, or one column, first or second column equals to 0, then determinant is equal to 0. Well, you just directly substitute. This is 1, 1. So this is 0. And this is 1, 2. So this is 0. Now in this case, this is 1, 2. So this is 0. And this is 2, 2. This is 0. So no matter what you do, you will get 0. So that's the quality number 1. Having the matrix a row or a column equal to 0, completely equal to 0, makes this matrix basically not acceptable for solving linear equations. And as you see, if you have a linear equation that we were talking about, let's say we consider this case. So 1, 1 and 1, 2 are equal to 0, which means we have this. And this is 5 and 6. If you imagine B1 is not equal to 0, obviously the whole system cannot have any solutions. Now if you imagine B1 is equal to 0, then the first equation actually doesn't contribute anything at all into the solution. And we have practically one equation with two variables, and it has infinite number of solutions. So that's what actually I meant. If determinant is equal to 0, then we can have either no solutions or infinite number of solutions. So that's the first one. The row or the column equal to 0 is a sufficient condition for determinant being equal to 0. OK. Next. Next is a little bit less demanding, but as bad. What I'm talking about is, for instance, you have one row proportional to another row, which means you have some kind of coefficient. If you multiply this row by this coefficient, you will have this, like what? Like this, for instance. So this is an example of one row being multiple of another row. What happens with determinant? Well, let's think about it. So this is 1, 1, which is equal to 2, 2, 1, right? So instead of a 1, 1, I will have certain multiplier by a 2, 1, right? Second row first column. Now, a 2, 2 remains as it is, minus. a 1, 2 is this one. And it's the same proportionality as a 2, 2, right? So it's k a 2, 2 times a 2, 1. 2, 1 is this one. As you see, they are equal to 1, 2, 1, 2, 2, 2, 2, k, and k, right? So the determinant is equal to 0. So proportionality is actually equivalent. Well, it's actually a sufficient condition for determinant being equal to 0. Proportionality of rows as well as columns will have exactly the same thing. Now, how is it reflected in our system of equations? Well, very simply. Here it is again. So we are replacing a 1, 1 with k times a 2, 1. So it's k times a 2, 1. And we replace a 1, 2 with k times a 2, 2. So what happens? Well, basically, I can say that b 1 is equal to k times, let's say, b 3, whatever it is. So I divide b 1 by k and get b 3. And now I can reduce it by k. And I have equation which is exactly the same on the right side, but different on the left. Now, if b 2 and b 3 are exactly the same, then one equation is just actually redundant, right? So we need only one equation with two variables. By the way, this is one, I'm sorry. One equation with two variables, which has infinite number of solutions. If b 1 and b 3 are different, then these three equations contradict each other, which means we don't have any solutions. And that's exactly what I'm talking about when I mentioned that the determinant is equal to 0, right? So proportionality is the same sufficient condition for the determinant being equal to 0. By the way, proportionality of roles in this particular case. Now, what if I'm talking about proportionality of the columns? So if I have, well, let me start from scratch. So if I have a proportionality of the columns, so let's say this column is proportional to this one. So this is k a 1 1, and this is k a 2 1. So what does it mean? Well, from the first equation, I have a 1 1 out. So I have b 1 over a 1 1 equals to x 1 plus k x 2. From the second equation, I have b 2 over a 2 1 equals to x 1 plus k x 2. So the right parts are exactly the same. Now, if the left parts are the same, then I have basically one extraneous equation. And from one equation, I can have multiple solutions, infinite number of solutions, because there are two variables here and one equation. If the left parts are not equal to each other, again, we have no solutions at all because they contradict each other. So it doesn't really matter whether it's columns proportional to each other or roles proportional to each other. Proportionality means that the determinant is equal to 0. Now, what if determinant is equal to 0? Let's go backwards. Does it signify that there is some proportionality? Well, in case coefficients are not equal to 0, let's think about it. If a 1 1, a 2 2 equals to a 1 2, a 2 1, right? If this is equal to 0, then I have this. Well, there are some coefficients which are not equal to 0, right? We have the grid. So let's just divide by, let's say you have a row, second row, which is a 2 1, a 2 2. Second row is not equal to 0. Then let's divide by this and you will have 1 1 over a 2 1 equals a 1 2 over a 2 2, right? If I divide both sides by a 2 2 times a 2 1, I will have this and I will have this, right? That's exactly the same thing, which means a proportionality, right? This is proportionality between a 1 1, a 1 2, a 2 1, and a 2 2. So this divided by this is equal to this divided by this. So the whole row is proportional to this row with some coefficient. As long as something is not equal to 0, right? So it goes both ways. In a non-zero case, proportionality and determinant equals to 0 are actually equivalent to each other, equivalent conditions. Now, what else is interesting about determinant? Well, here it is. Let's go to geometry now. And let's have a plane with two vectors. So you and we are vectors within this x y plane, horizontal plane. Now, what would be their vector product? Well, vector product, as we know, in the coordinate system, vector product of two vectors can be expressed as some functions of coordinates. So let's say vector u has coordinates u 1, u 2, u 3. And by the way, since vector u is completely within the plane x y plane, the z coordinate, the u 3, is equal to actually 0. So it's actually u 1, u 2, 0. Now, vector d, which has coordinates v 1, v 2, v 3, also has v 3 equals to 0, right? Because both vectors are within the plane, x y plane. Now, if w is their cross product, vector product, then it has coordinates w 1, w 2, w 3. Now, let's think about. Both vectors are within this horizontal plane. And we know that vector product is perpendicular to both, which means vector product should go upwards along the z-axis, along the z-axis. Well, but in any case, we have basically derived the general formula. If you have these coordinates of two vectors, then these coordinates are basically calculated using the following formulas. u 2 v 3 minus u 3 v 2, w 2 equals, it's all cyclically, it has cyclically changed. u 3 v 1 minus u 1 v 3. And v w 3 is equal to u 1 v 2 minus u 2 v 1. Now, since v 3 is equal to 0 and u 3 is equal to 0, v 3 and u 3, so this is 0. u 3 and v 3 is equal to 0, so this is 0. So basically, the result, the vector product, has w 1 and w 2, which are x and y coordinates, equal to 0, which is to be expected, right? Since the vector is completely collinear with the z-axis, it has x and y coordinates equal to 0. And what is the third coordinate? Well, that's quite interesting. Look, if you will put the coordinates of this vector in the matrix where the first vector occupies the first row and the second vector occupies the second row, so I don't really need the third constant, it's equal to 0. So within this plane, u has coordinates u 1, u 2, and v has a coordinate v 1, v 2. So if I put it in this matrix, what would be the determinant of this matrix? Well, u 1 times v 2 minus u 2 times v 1, which is this one. Isn't that interesting? So the vector product has two coordinates equal to 0. The z-coordinate is equal to determinant, right, the matrix which consists of these coefficients. And now what was the geometric representation, geometric meaning of the vector product? If you remember the absolute value of vector w, it's area of parallelogram, this one. Which is built on both vectors u and v. So the area of this parallelogram is equal to the length of the vector w. And the length of the vector w basically is absolute value of w 3, because w 1 and w 2 are equal to 0. So it's vector which is stretched along the z-axis. So basically the absolute value of the z-coordinate is its length. So what we came up with is, we came up with a very interesting conclusion. If I have two vectors on the plane with coordinates u 1, u 2 and v 1, v 2, now my plane is the surface of this whiteboard. Then the area of this parallelogram is equal to, so this is x, this is y, so this is coordinates u 1, u 2, v 1, v 2. So the area of this parallelogram is equal to a determinant, for actual absolute value of determinant of the matrix, which,