Formulation of the problem
The task requires the user to become familiar with the basic concepts of numerical methods, such as the determinant and inverse matrix, and various ways of calculating them. This theoretical report first introduces the basic concepts and definitions in simple and accessible language, on the basis of which further research is carried out. The user may not have special knowledge in the field of numerical methods and linear algebra, but can easily use the results of this work. For clarity, a program for calculating the determinant of a matrix using several methods, written in the C++ programming language, is given. The program is used as a laboratory stand for creating illustrations for the report. A study of methods for solving systems of linear algebraic equations is also being conducted. The uselessness of calculating the inverse matrix is proven, so the work provides more optimal ways to solve equations without calculating it. It explains why there are so many different methods for calculating determinants and inverse matrices and discusses their shortcomings. Errors in calculating the determinant are also considered and the achieved accuracy is assessed. In addition to Russian terms, the work also uses their English equivalents to understand under what names to look for numerical procedures in libraries and what their parameters mean.
Basic definitions and simplest properties
Determinant
Let us introduce the definition of the determinant of a square matrix of any order. This definition will be recurrent, that is, in order to establish what the determinant of the order matrix is, you need to already know what the determinant of the order matrix is. Note also that the determinant exists only for square matrices.
We will denote the determinant of a square matrix by or det.
Definition 1. Determinant square matrix 
second order number is called 
.
Determinant 
square matrix of order , is called the number 
where is the determinant of the order matrix obtained from the matrix by deleting the first row and column with number .
For clarity, let’s write down how you can calculate the determinant of a fourth-order matrix:
Comment. The actual calculation of determinants for matrices above third order based on the definition is used in exceptional cases. Typically, the calculation is carried out using other algorithms, which will be discussed later and which require less computational work.
Comment. In Definition 1, it would be more accurate to say that the determinant is a function defined on the set of square matrices of order and taking values in the set of numbers.
Comment. In the literature, instead of the term “determinant”, the term “determinant” is also used, which has the same meaning. From the word “determinant” the designation det appeared.
Let us consider some properties of determinants, which we will formulate in the form of statements.
Statement 1. When transposing a matrix, the determinant does not change, that is, .
Statement 2. The determinant of the product of square matrices is equal to the product of the determinants of the factors, that is.
Statement 3. If two rows in a matrix are swapped, its determinant will change sign.
Statement 4. If a matrix has two identical rows, then its determinant is zero.
In the future, we will need to add strings and multiply a string by a number. We will perform these actions on rows (columns) in the same way as actions on row matrices (column matrices), that is, element by element. The result will be a row (column), which, as a rule, does not coincide with the rows of the original matrix. If there are operations of adding rows (columns) and multiplying them by a number, we can also talk about linear combinations of rows (columns), that is, sums with numerical coefficients.
Statement 5. If a row of a matrix is multiplied by a number, then its determinant will be multiplied by this number.
Statement 6. If a matrix contains a zero row, then its determinant is zero.
Statement 7. If one of the rows of the matrix is equal to another, multiplied by a number (the rows are proportional), then the determinant of the matrix is equal to zero.
Statement 8. Let the i-th row in the matrix have the form . Then , where the matrix is obtained from the matrix by replacing the i-th row with the row , and the matrix is obtained by replacing the i-th row with the row .
Statement 9. If you add another row to one of the matrix rows, multiplied by a number, then the determinant of the matrix will not change.
Statement 10. If one of the rows of a matrix is a linear combination of its other rows, then the determinant of the matrix is equal to zero.
Definition 2. Algebraic complement to a matrix element is a number equal to , where is the determinant of the matrix obtained from the matrix by deleting the i-th row and j-th column. The algebraic complement of a matrix element is denoted by .
Example. Let 
. Then 
Comment. Using algebraic additions, the definition of 1 determinant can be written as follows: 
Statement 11. Expansion of the determinant in an arbitrary string.
The formula for the determinant of the matrix is 
Example. Calculate 
.
Solution. Let's use the expansion along the third line, this is more profitable, since in the third line two of the three numbers are zeros. We get
Statement 12. For a square matrix of order at, the relation holds: 
.
Statement 13. All properties of the determinant formulated for rows (statements 1 - 11) are also valid for columns, in particular, the decomposition of the determinant in the j-th column is valid 
and equality 
at .
Statement 14. The determinant of a triangular matrix is equal to the product of the elements of its main diagonal.
Consequence. The determinant of the identity matrix is equal to one, .
Conclusion. The properties listed above make it possible to find determinants of matrices of sufficiently high orders with a relatively small amount of calculations. The calculation algorithm is as follows.
Algorithm for creating zeros in a column. Suppose we need to calculate the order determinant. If , then swap the first line and any other line in which the first element is not zero. As a result, the determinant , will be equal to the determinant of the new matrix with the opposite sign. If the first element of each row is equal to zero, then the matrix has a zero column and, according to statements 1, 13, its determinant is equal to zero.
So, we believe that already in the original matrix . We leave the first line unchanged. Add to the second line the first line multiplied by the number . Then the first element of the second line will be equal to 
.
We denote the remaining elements of the new second row by , . The determinant of the new matrix according to statement 9 is equal to . Multiply the first line by a number and add it to the third. The first element of the new third line will be equal to 
We denote the remaining elements of the new third row by , . The determinant of the new matrix according to statement 9 is equal to .
We will continue the process of obtaining zeros instead of the first elements of lines. Finally, multiply the first line by a number and add it to the last line. The result is a matrix, let’s denote it , which has the form 
and . To calculate the determinant of the matrix, we use expansion in the first column
Since then 
On the right side is the determinant of the order matrix. We apply the same algorithm to it, and calculating the determinant of the matrix will be reduced to calculating the determinant of the order matrix. We repeat the process until we reach the second-order determinant, which is calculated by definition.
If the matrix does not have any specific properties, then it is not possible to significantly reduce the amount of calculations compared to the proposed algorithm. Another good aspect of this algorithm is that it is easy to use it to create a computer program for calculating determinants of matrices of large orders. Standard programs for calculating determinants use this algorithm with minor changes related to minimizing the influence of rounding errors and input data errors in computer calculations.
Example. Compute determinant of matrix 
.
Solution. We leave the first line unchanged. To the second line we add the first, multiplied by the number:
The determinant does not change. To the third line we add the first, multiplied by the number:
The determinant does not change. To the fourth line we add the first, multiplied by the number:
The determinant does not change. As a result we get 
Using the same algorithm, we calculate the determinant of the matrix of order 3, located on the right. We leave the first line unchanged, add the first line multiplied by the number to the second line 
:
To the third line we add the first, multiplied by the number 
:
As a result we get
Answer. .
Comment. Although fractions were used in the calculations, the result turned out to be a whole number. Indeed, using the properties of determinants and the fact that the original numbers are integers, operations with fractions could be avoided. But in engineering practice, numbers are extremely rarely integers. Therefore, as a rule, the elements of the determinant will be decimal fractions and it is inappropriate to use any tricks to simplify the calculations.
inverse matrix
Definition 3. The matrix is called inverse matrix for a square matrix, if .
From the definition it follows that the inverse matrix will be a square matrix of the same order as the matrix (otherwise one of the products or would not be defined).
The inverse of a matrix is denoted by . Thus, if exists, then .
From the definition of an inverse matrix it follows that the matrix is the inverse of the matrix, that is, . We can say about matrices that they are inverse to each other or mutually inverse.
If the determinant of a matrix is zero, then its inverse does not exist.
Since to find the inverse matrix it is important whether the determinant of the matrix is equal to zero or not, we introduce the following definitions.
Definition 4. Let's call the square matrix degenerate or special matrix, if non-degenerate or non-singular matrix, If .
Statement. If the inverse matrix exists, then it is unique.
Statement. If a square matrix is non-singular, then its inverse exists and 
(1) where are algebraic complements to the elements.
Theorem. An inverse matrix for a square matrix exists if and only if the matrix is non-singular, the inverse matrix is unique, and formula (1) is valid.
Comment. Particular attention should be paid to the places occupied by algebraic additions in the inverse matrix formula: the first index shows the number column, and the second is the number lines, in which you need to write the calculated algebraic addition.
Example. 
.
Solution. Finding the determinant
Since , then the matrix is non-degenerate, and its inverse exists. Finding algebraic complements:
We compose the inverse matrix, placing the found algebraic complements so that the first index corresponds to the column, and the second to the row: 
(2)
The resulting matrix (2) serves as the answer to the problem.
Comment. In the previous example, it would be more accurate to write the answer like this:
(3)
However, notation (2) is more compact and it is more convenient to carry out further calculations with it, if required. Therefore, writing the answer in the form (2) is preferable if the matrix elements are integers. And vice versa, if the elements of the matrix are decimal fractions, then it is better to write the inverse matrix without a factor in front.
Comment. When finding the inverse matrix, you have to perform quite a lot of calculations and the rule for arranging algebraic additions in the final matrix is unusual. Therefore, there is a high probability of error. To avoid errors, you should check: calculate the product of the original matrix and the final matrix in one order or another. If the result is an identity matrix, then the inverse matrix has been found correctly. Otherwise, you need to look for an error.
Example. Find the inverse of a matrix 
.
Solution.
- exists.
Answer: 
.
Conclusion. Finding the inverse matrix using formula (1) requires too many calculations. For matrices of fourth order and higher, this is unacceptable. The actual algorithm for finding the inverse matrix will be given later.
Calculating the determinant and inverse matrix using the Gaussian method
The Gaussian method can be used to find the determinant and inverse matrix.
Namely, the determinant of the matrix is equal to det.
The inverse matrix is found by solving systems of linear equations using the Gaussian elimination method:
Where is the j-th column of the identity matrix, is the desired vector.
The resulting solution vectors obviously form columns of the matrix, since .
Formulas for the determinant
1.
If the matrix is non-singular, then and (product of leading elements).
Often in universities we come across problems in higher mathematics in which it is necessary calculate the determinant of a matrix. By the way, the determinant can only be in square matrices. Below we will consider the basic definitions, what properties the determinant has and how to calculate it correctly. We will also show a detailed solution using examples.
What is the determinant of a matrix: calculating the determinant using the definition
Matrix determinant
Second order is a number.
The determinant of a matrix is denoted – (short for the Latin name for determinants), or .
If:, then it turns out
Let us recall a few more auxiliary definitions:
Definition
An ordered set of numbers that consists of elements is called a permutation of order.
For a set that contains elements there is a factorial (n), which is always denoted by an exclamation mark: . The permutations differ from each other only in the order in which they appear. To make it clearer, let's give an example:
Consider a set of three elements (3, 6, 7). There are 6 permutations in total, since .:
Definition
An inversion in a permutation of order is an ordered set of numbers (it is also called a bijection), where two of them form a kind of disorder. This is when the larger number in a given permutation is located to the left of the smaller number.
Above we looked at an example with the inversion of a permutation, where there were numbers . So, let’s take the second line, where judging by these numbers it turns out that , a , since the second element is greater than the third element. Let's take for comparison the sixth line, where the numbers are located: . There are three pairs here: , and , since title="Rendered by QuickLaTeX.com" height="13" width="42" style="vertical-align: 0px;">; , так как title="Rendered by QuickLaTeX.com" height="13" width="42" style="vertical-align: 0px;">; , – title="Rendered by QuickLaTeX.com" height="12" width="43" style="vertical-align: 0px;">.!}
We will not study the inversion itself, but permutations will be very useful to us in further consideration of the topic.
Definition
Determinant of matrix x – number:
is a permutation of numbers from 1 to an infinite number, and is the number of inversions in the permutation. Thus, the determinant includes terms that are called “terms of the determinant”.
You can calculate the determinant of a matrix of second, third, and even fourth order. Also worth mentioning:
Definition
The determinant of a matrix is the number that equals
To understand this formula, let us describe it in more detail. The determinant of a square matrix x is a sum that contains terms, and each term is the product of a certain number of matrix elements. Moreover, in each product there is an element from each row and each column of the matrix.
It may appear before a certain term if the matrix elements in the product are in order (by row number), and the number of inversions in the permutation of many column numbers is odd.
It was mentioned above that the determinant of a matrix is denoted by or, that is, the determinant is often called a determinant.
So, let's return to the formula:
From the formula it is clear that the determinant of a first-order matrix is an element of the same matrix.
Calculation of the determinant of a second-order matrix
Most often in practice, the determinant of a matrix is solved using methods of the second, third, and less often, fourth order. Let's look at how the determinant of a second-order matrix is calculated:
In a second-order matrix, it follows that the factorial is . Before you apply the formula
It is necessary to determine what data we obtain:
2. permutations of sets: and ;
3. number of inversions in the permutation : and , since title="Rendered by QuickLaTeX.com" height="13" width="42" style="vertical-align: -1px;">;!}
4. corresponding works: and.
It turns out:
Based on the above, we obtain a formula for calculating the determinant of a second-order square matrix, that is, x:
Let's look at a specific example of how to calculate the determinant of a second-order square matrix:
Example
Task
Calculate the determinant of the matrix x:
Solution
So, we get , , , .
To solve, you need to use the previously discussed formula:
We substitute the numbers from the example and find:
Answer
Second order matrix determinant = .
Calculation of the determinant of a third-order matrix: example and solution using the formula
Definition
The determinant of a third-order matrix is a number obtained from nine given numbers arranged in a square table,
The third order determinant is found in almost the same way as the second order determinant. The only difference is in the formula. Therefore, if you understand the formula well, then there will be no problems with the solution.
Consider a third-order square matrix *:
Based on this matrix, we understand that, accordingly, factorial = , which means that the total permutations are
To apply the formula correctly, you need to find the data:
So, the total permutations of the set are:
The number of inversions in the permutation is , and the corresponding products = ;
Number of inversions in permutation title="Rendered by QuickLaTeX.com" height="18" width="65" style="vertical-align: -4px;">, соответствующие произведения = ;!}
Inversions in permutation title="Rendered by QuickLaTeX.com" height="18" width="65" style="vertical-align: -4px;"> ;!}
. ; inversions in permutation title="Rendered by QuickLaTeX.com" height="18" width="118" style="vertical-align: -4px;">, соответствующие произведение = !}
. ; inversions in permutation title="Rendered by QuickLaTeX.com" height="18" width="118" style="vertical-align: -4px;">, соответствующие произведение = !}
. ; inversions in permutation title="Rendered by QuickLaTeX.com" height="18" width="171" style="vertical-align: -4px;">, соответствующие произведение = .!}
Now we get:
Thus, we have a formula for calculating the determinant of a matrix of order x:
Finding a third-order matrix using the triangle rule (Sarrus rule)
As mentioned above, the elements of the 3rd order determinant are located in three rows and three columns. If you enter the designation of the general element, then the first element denotes the row number, and the second element from the indices denotes the column number. There is a main (elements) and secondary (elements) diagonals of the determinant. The terms on the right side are called terms of the determinant).
It can be seen that each term of the determinant is in the diagram with only one element in each row and each column.
You can calculate the determinant using the rectangle rule, which is depicted in the form of a diagram. The terms of the determinant from the elements of the main diagonal are highlighted in red, as well as the terms from the elements that are at the vertex of triangles that have one side parallel to the main diagonal (left diagram), taken with the sign .
Terms with blue arrows from elements of the side diagonal, as well as from elements that are at the vertices of triangles that have sides parallel to the side diagonal (right diagram) are taken with the sign.
Using the following example, we will learn how to calculate the determinant of a third-order square matrix.
Example
Task
Calculate the determinant of a third-order matrix:
Solution
In this example:
We calculate the determinant using the formula or scheme discussed above:
Answer
Determinant of a third-order matrix =
Basic properties of determinants of a third-order matrix
Based on the previous definitions and formulas, let's consider the main properties of the matrix determinant.
1.
The size of the determinant will not change when replacing the corresponding rows and columns (such a replacement is called transposition).
Using an example, we will make sure that the determinant of the matrix is equal to the determinant of the transposed matrix:
Let us recall the formula for calculating the determinant:
Transpose the matrix:
We calculate the determinant of the transposed matrix:
We have verified that the determinant of the transported matrix is equal to the original matrix, which indicates the correct solution.
2.
The sign of the determinant will change to the opposite if any two of its columns or two rows are swapped.
Let's look at an example:
Given two third-order matrices (x):
It is necessary to show that the determinants of these matrices are opposite.
Solution
The rows in the matrix and in the matrix have changed (the third from the first, and from the first to the third). According to the second property, the determinants of two matrices must differ in sign. That is, one matrix has a positive sign, and the second one has a negative sign. Let's check this property by using the formula to calculate the determinant.
The property is true because .
3.
A determinant is equal to zero if it has the same corresponding elements in two rows (columns). Let the determinant have identical elements of the first and second columns:
By swapping identical columns, we, according to Property 2, obtain a new determinant: = . On the other hand, the new determinant coincides with the original one, since the elements have the same answers, that is, = . From these equalities we get: = .
4.
The determinant is equal to zero if all elements of one row (column) are zero. This statement emerges from the fact that each term of the determinant according to formula (1) has one, and only one element from each row (column), which has only zeros.
Let's look at an example:
Let us show that the determinant of the matrix is equal to zero:
Our matrix has two identical columns (second and third), therefore, based on this property, the determinant must be equal to zero. Let's check:
Indeed, the determinant of a matrix with two identical columns is equal to zero.
5.
The common factor of the elements of the first row (column) can be taken out of the determinant sign:
6.
If the elements of one row or one column of a determinant are proportional to the corresponding elements of the second row (column), then such a determinant is equal to zero.
Indeed, following property 5, the coefficient of proportionality can be taken out of the sign of the determinant, and then property 3 can be used.
7.
If each of the elements of the rows (columns) of the determinant is the sum of two terms, then this determinant can be presented as the sum of the corresponding determinants:
To check, it is enough to write in expanded form according to (1) the determinant that is on the left side of the equality, then separately group the terms that contain the elements and . Each of the resulting groups of terms will be, respectively, the first and second determinant on the right side of the equality.
8.
The definition values will not change if the corresponding elements of the second row (column) are added to an element of one row or column, multiplied by the same number:
This equality is obtained based on properties 6 and 7.
9.
The determinant of the matrix, , is equal to the sum of the products of the elements of any row or column and their algebraic complements.
Here by means the algebraic complement of a matrix element. Using this property, you can calculate not only third-order matrices, but also matrices of higher orders (x or x). In other words, this is a recurrent formula that is needed in order to calculate the determinant of a matrix of any order. Remember it, as it is often used in practice.
It is worth saying that using the ninth property it is possible to calculate the determinants of matrices not only of the fourth order, but also of higher orders. However, in this case you need to perform a lot of computational operations and be careful, since the slightest error in the signs will lead to an incorrect decision. It is most convenient to solve matrices of higher orders using the Gaussian method, and we will talk about this later.
10.
The determinant of the product of matrices of the same order is equal to the product of their determinants.
Let's look at an example:
Example
Task
Make sure that the determinant of two matrices and is equal to the product of their determinants. Two matrices are given:
Solution
First, we find the product of the determinants of two matrices and .
Now let's multiply both matrices and thus calculate the determinant:
Answer
We made sure that
Calculating the determinant of a matrix using the Gaussian method
Matrix determinant updated: November 22, 2019 by: Scientific Articles.Ru
Exercise. Calculate the determinant by decomposing it into elements of some row or some column.
Solution. Let us first perform elementary transformations on the rows of the determinant, making as many zeros as possible either in the row or in the column. To do this, first subtract nine thirds from the first line, five thirds from the second, and three thirds from the fourth, we get:
Let us decompose the resulting determinant into the elements of the first column:
We will also expand the resulting third-order determinant into the elements of the row and column, having previously obtained zeros, for example, in the first column. To do this, subtract the second two lines from the first line, and the second line from the third:
Answer. 
12. Slough 3rd order
1. Triangle rule
Schematically, this rule can be depicted as follows:
The product of elements in the first determinant that are connected by straight lines is taken with a plus sign; similarly, for the second determinant, the corresponding products are taken with a minus sign, i.e.
2. Sarrus' rule
To the right of the determinant, add the first two columns and take the products of elements on the main diagonal and on the diagonals parallel to it with a plus sign; and the products of the elements of the secondary diagonal and the diagonals parallel to it, with a minus sign:
3. Expansion of the determinant in a row or column
The determinant is equal to the sum of the products of the elements of the row of the determinant and their algebraic complements. Usually the row/column that contains zeros is selected. The row or column along which the decomposition is carried out will be indicated by an arrow.
Exercise. Expanding along the first row, calculate the determinant
Solution.
Answer. 
4. Reducing the determinant to triangular form
Using elementary transformations over rows or columns, the determinant is reduced to a triangular form and then its value, according to the properties of the determinant, is equal to the product of the elements on the main diagonal.
Example
Exercise. Compute determinant 
bringing it to a triangular form.
Solution. First we make zeros in the first column under the main diagonal. All transformations will be easier to perform if the element is equal to 1. To do this, we will swap the first and second columns of the determinant, which, according to the properties of the determinant, will cause it to change its sign to the opposite:
Next, we get zeros in the second column in place of the elements under the main diagonal. Again, if the diagonal element is equal to , then the calculations will be simpler. To do this, swap the second and third lines (and at the same time change to the opposite sign of the determinant):
Next, we make zeros in the second column under the main diagonal, to do this we proceed as follows: we add three second rows to the third row, and two second rows to the fourth, we get:
Next, from the third line we take (-10) out of the determinant and make zeros in the third column under the main diagonal, and to do this we add the third to the last line:
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