Математика. Материалы курса

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At this lecture, we will introduce the concept of the determinant of a square matrix. This concept is used in the study of systems of linear equations. First, we define what is meant by the determinant of a 2-by-2 matrix, i.e. we will consider a simple case. Let’s start with the system of two linear equations with two variables. Let’s assume that all coefficients for variables are non-zero and perform the following transformation. We multiply the first equation by c and the second one by a. After that, the coefficients for the variable x are equalized, and we make a subtraction, we subtract the second from the second equation, and we get a step system. We assume that the coefficient for the variable y is not zero, the variable y can be expressed, and if we do not have a consistent non-zero equation in the step system, the system has exactly one solution. So, y is expressed. The variable x can be expressed in a similar way if we take the original system and perform the specified operations, then subtract the second from the first equation and again, under our assumption, express the variable x. Note that even if some of the numbers a and c, b and d that we multiplied by are equal to zero, the formulas still hold. The main thing is that the expression ad - bc is not equal to zero, but if this expression turns to zero, in this case, the system either has no solutions, or there are infinitely many solutions. And so we summarize the preliminary result, if the system has one solution, it can be found by the derived formulas. These formulas are difficult to remember, but the expression that is present in them is determined by some general rule from the corresponding matrices. Let us first consider the expression in the denominator (ad - bc). This expression is called the second-order matrix determinant. This matrix is obtained from the coefficients for variables. If we consider the elements a and d, and multiply them, and then subtract the product of the elements bc, we get the number, which is in the denominator of the derived formulas. In this case, the elements ad and bc are located on the diagonal of our 2 by 2 table, while the diagonal ad is called the main diagonal, and the diagonal bc is called the secondary diagonal. Let’s denote this number with the letter Delta and look at other expressions. So, in the numerator of the fraction, there is a number for the variable x which is obtained by the same rule, only for a different matrix. To construct it, we put the column of free members k and r instead of the first column. At first, we had a and c, now instead of the column a and c, there is a column of the parameters k and r. This number is found by the same rule. We multiply the elements of the main diagonal kd, subtract the product of the elements of the secondary diagonal br. And similarly, the numerator (аr-kc) is obtained by the same rule from the determinant of the matrix, which is formed after we write the column kr instead of the second column, which corresponds to the variable y. So, the derived formulas can be written in short as follows (see the video). The solution of a system of equations using these formulas is called Cramer’s rule. In this case, this rule can be applied not only for the system that we have considered, but also for some more general situation when the main matrix of the system has an arbitrary n-th order, i.e. it is square and its determinant Delta is not equal to zero. But first we need to extend the concept of a determinant for an arbitrary square matrix of the n-th order. So, we considered how the determinant of the second-order matrix is calculated. Let’s give an example (see the video). So, 2*4 and subtract 3*-1, we get the number 11. If n is equal to one, i.e. we have a first-order matrix, in this case, it degenerates into one element, the determinant of such a matrix is assumed to be equal to this element. In order to avoid ambiguity, we can first write a matrix composed of a single element a by enclosing this element in the parentheses, note that it is the matrix that is being considered. And then we write down its determinant, which is equal to a. However, to make the entries simple, the parentheses are usually omitted and only vertical brackets are used to indicate the determinant. Now, let’s consider the case when n=3, i.e. there is a third-order matrix. The determinant of such a matrix is found by the specified formula (see the video). Again, we get that the formula itself is difficult to understand, so various schemes are used to use it. Let’s look at the first three terms, each of them is a product of three elements of the matrix. In this case, the first term is the product of the elements standing on the main diagonal, and the other two terms are the product of three elements that are located at the vertices of conditional triangles. Let’s show the triangles on the slide. Now, let’s look at the sum in the parentheses. If the parentheses are opened, a minus sign appears before each term, and again the first term is the product of the elements located on the other diagonal of the сyu, i.e. on the secondary diagonal. And then two products that are represented by the elements located at the vertices of other conditional triangles. In order not to remember how to get triangles, we can use a different rule, a different scheme. We duplicate the first two columns to the right of the original matrix. In this case, the first multiplier is an element that is located, as we already know, on the main diagonal, and the other two products are obtained by multiplying the numbers that are on the conditional straight lines parallel to the main diagonal. For another sum, we have a similar situation. First, we take the elements on the secondary diagonal, and then consider the product of the numbers on the lines that are parallel to this diagonal. Let’s take the following example (see the video). We add the first two columns to the right and find three products. Each product is highlighted in its own color. We take all these three products with a plus sign and add them together. Next, we find the terms before which the minus sign appears. After that, we make calculations and get the answer. In this case, the determinant is equal to the number minus 24. And for the cases when n is equal to 1, 2 and 3, we have formulas. Now, let’s consider how to introduce the concept of a determinant for an arbitrary n-order matrix. In this case, we will need to consider a method that allows us to get an n-1 order matrix from an n-th order matrix, i.e. next lower order. To do this, we will select an element of this matrix and consider another matrix obtained from the original one by crossing out the row and column in which this element is located. So, using this idea, let’s write down the formula that allows us to calculate an arbitrary determinant (see the video). This formula is called the expansion of the determinant along the first row. So, we consider the first row and make up the next sum, each summand of this sum is a product of the row element and the so-called algebraic complement of this element denoted by A. The element a₁₁ is multiplied by the complement of A₁₁, after that we take the element a₁₂, multiply by its algebraic complement, and so on. What is an algebraic complement? This is the determinant of the matrix, which is obtained by crossing out the row and column in which the corresponding element is located, and this number is multiplied by another coefficient -1^i+j. In fact, this coefficient is equal to either one or minus one, depending on the parity of the indicator. Thus, if the sum of the indices is odd, we get a minus sign before the determinant, but if the sum of the indices is even, the unit multiplier does not change the existing determinant in any way. This definition reduces the determinant of an n-order matrix to the calculation of a lower-order determinant. Here is an example (see the video). We calculate the third-order determinant using the first-line expansion. We fix the first line, and multiply each element of the i-th line by the corresponding complements. Now, we calculate these complements according to the specified rule. So, A₁₁ is obtained by crossing out the first row and the first column. Here, the sum of the indices is even, so the determinant is equal to 6. It will be the value of complements of the element 11. Next, we calculate the complement of element12, after crossing out, we obtain the matrix (see the video) and its determinant is equal to three. However, given the minus sign that appears, we get the number -3. Similarly, we calculate the third complement. We cross out the first row and the third column and make calculations, we get the number -6. We substitute the obtained results into our expansion and after calculations we get 9. Now, we consider the properties that the determinant of a matrix of an arbitrary order obeys. To illustrate it, we will give simple examples. First, if the rows in a square matrix are mutually replaced by columns, its determinant will not change. In other words, transposing does not change the determinant of the square matrix. For example (see the video), both determinants are equal to the same number. Thus, from the point of view of determinants, the rows and columns of the matrix are equal. Second, the determinant can be expanded into any row. The formulas take the same form, i.e. we get the sum in which each term is a product of the elements of the row and its algebraic complement. A number of consequences can be deduced from this. First, if you multiply an arbitrary row of the matrix by the number k, we multiply the determinant of this matrix by this number. Indeed, after we expand the determinant along the given row, in each summand, we will have a common multiplier, after which we put it out of the parenthesis. To illustrate it, we consider this example (see the video). Second, if the determinant has a zero row, it is equal to zero. Again, we expand the determinant along this row and get the sum in which all the terms are equal to zero, which means that the entire sum is zero. The third property is that if you add another row multiplied by an arbitrary number to one row, the determinant will not change. The subtraction is made in a similar way. From this, we can deduce the following statement. If we get two equal rows in the determinant, we can immediately say that it is equal to zero. Indeed, we subtract an equal row from the row, get a zero row, and by virtue of the above, we get the value 0. Similarly, if the determinant has 2 proportional rows, it is also zero. In this example, the first row is equal to three, so it is zero without any calculations. So, there is another property. Swapping two rows changes the sign of the determinant. Note, first, that all the properties discussed above assume operations with rows. However, these properties will be preserved if we perform the same transformations on columns, due to the fact that rows and columns are equal, which follows from property 1. Second, the use of these properties makes it easier to calculate the determinant. As a rule, it works according to the following scheme. Using elementary transformations, we make sure that there are as many zeros as possible in a certain row, and then expand the determinant along this row. We can also try to bring the matrix to a step-by-step form, if suddenly we have a zero row, the determinant will be zero, but if zero rows do not appear, we will get the so-called triangular matrix. This is a matrix that has all the elements below the main diagonal equal to zero. There is such a property that the determinant of a triangular matrix is equal to the product of the elements standing on the main diagonal. For example, like this (see the video). Note that if we remove the zeros that are below the diagonal, our matrix will take the form of a right triangle, which is why it is called triangular. We will consider other examples of using these properties when calculating the determinant at our practical lesson.