Lecture 6.

Matrians

6.1. Basic concepts

Definition 1.The matrix is \u200b\u200bcalled a rectangular table of numbers.

To designate the matrix, round brackets or dual vertical lines are used:

The numbers constituting the matrix are called it elements, element matrians located in It Line I. - Column.

Numbers and (The number of rows and columns of the matrix) are called its orders.

They also say that - Matrix size
.

If a
, the matrix called square.

An designation is also used for brief record
(or
) and then it is indicated in what limits they change and , eg,
,
,
. (Recording is read like this: matrix with elements ,changes from before ,- OT before .)

Among square matrices diagonal matriceswho have all elements with unequal indexes (
) are zero:

.

We will say that elements
located on the main diagonal.

Diagonal matrix of view

called singlematrix.

Later will meet matrices of the species

and
,

which are called triangularmatrians, as well as matrices consisting of one column:

and one line:

(column Matrix and Matrix String).

Matrix, all the elements of which are equal to zero, called null.

6.2. Decreased order n.

Let a square matrix of order be given :

. (6.1)

Make all sorts of works matrix elements located in different lines and different columns, i.e. Works of type

. (6.2)

The number of works of the form (6.2) is equal (We will take this fact without proof).

We will consider all these works by members of the policy determinant corresponding to the matrix (6.1).

The second indices of multipliers in (6.2) constitute the permutation of the first natural numbers
.

Say that numbers and in the permutation make up inversion, if a
, and in the permutation located before .

Example 1.In the permutation of six numbers
numbers and ,and ,and ,and ,and make up inversion.

The permutation is called evenif the number of inversions in it is even, and oddIf the number of inversions in it is odd.

Example 2.Perestanovka
- odd, and permutation
- even ( inversions).

Definition 2.Determinant order , The corresponding matrix(6.1), is called an algebraic amount members, Compiled as follows: Members of the determinant are all sorts of works matrix elements, taken one of each row and each column, and the term is taken with a sign"+", If the set of second indices is an even permutation of numbers
, And with friends"–", If odd.

Denote by the determinant of the matrix (6.1) accepted as:

.

Comment. Definition 2 for
and
leads to us already familiar to us by the determinants of the 2nd and 3rd order:

,

Transposingaround the main diagonal of the matrix called transition to the matrix
for which the strings of the matrix are columns, and columns - lines:

.

We say that the determinant
received by transposition of the determinant .

Properties of the determinant of order n:

1.
(The determinant does not change during transposition around the main diagonal).

2. If one of the rows of the determinant consists of zeros, the determinant equal to zero..

3. From the permutation of two lines, the determinant changes only a sign.

4. The determinant containing two identical lines is zero.

5. If all the elements of a certain line of the determinant multiply by the number , the determinant will multiply on .

6. The determinant containing two proportional lines is zero.

7. If all items - Lines of the determinant are represented as a sum
then the determinant is equal to the sum of two determinants who have all the lines except -y, the same as in the source determinant, and -Ith line in one determined consists of , and in the other - from .

Definition 3.-In the string of the determinant is called a linear combination of his remaining lines, If such, What, multiplying Line on , and then folding all lines, Besides -y., Receive - Line.

8. If one of the rows of the determinant is a linear combination of its remaining rows, the determinant is zero.

9. The determinant will not change if it is added to the elements of one of its lines to add the corresponding elements other multiplied by the same number.

Comment. We formulated the properties of the determinant for strings. By virtue of properties 1 (
) They are valid for columns.

All preferred properties were proven to practical activities for
; For arbitrary we will take them without proof.

If in the definition order choose an item and delete the column and string at the intersection of which is located The remaining lines and columns form the determinant of order
called minordeterminant corresponding to the element .

Example 3.In the identifier

minor element
is determined
.

Definition 4.Algebraic supplement element determinant called his minor, Multiplicated by
, Where - Row number, - Column number, in which the selected item is located .

Example 4. In the identifier

algebraic supplement
.

Theorem 1 (on line decomposition).The determinant is equal to the amount of works of all elements of any row on their algebraic additions.

Theorem 1 allows you to reduce the calculation of the determinant of order to calculate order determinants
.

Example 5.. Calculate the fourth order determinant:

.

We use Theorem 1 and decompose the determinant on the 4th line:

Comment. You can first simplify the determinant, using the property 9, and then use theorem 1. Then the calculation of the order determinant cut down to calculate just oneorder determinant
.

Example 6.Calculate

.

I add the first column to the second and the first column multiplied by (
), to the third, as a result we get

.

Now we will apply theorem 1 and decompose on the last line:

,

the calculation of the 4th order determinant was reduced to the calculation of just one determinant of the 3rd order.

,

the calculation of the third order determinant has begun to calculate only one second order determinant.

Example 7.Calculate the determinant of order :

.

First string add to the second, third, etc. Line. Come to the determinant

.

The determinant of the triangular species is obtained.

Apply
once theorem 1 (decompose on the first column) and we get

.

Comment. The determinant of the triangular species is equal to the product of the elements of the main diagonal.

6.3. Basic operations on matrices

Definition 5.Two matrices
,
,
, and
,
,
, Let's call equal if
.

Short record:
.

Thus, two matrices are considered equal if they have the same orders and their corresponding elements are equal.

Definition 6.Sum of two matrices
,
,
, and
,
,
, This matrix is \u200b\u200bcalled
,
,
, what
.

In other words, only the matrices of the same orders can be folding, and the addition is carried out alternately.

Example 8.Find the sum of the matrix

and
.

In accordance with definition 6 we will find

.

The final of the arrangement of matrices applies to the amount of any finite number of terms.

Definition 7.The work of the matrix
,
,
, for real number this matrix is \u200b\u200bcalled
,
,
, for which
.

In other words, to multiply the matrix to the number, you need to multiply all its elements on this number and leave the obtained works in the previous places.

Example 9.Find a linear combination
matrix

and
.

Using the definition 7, we get

,
,

.

Properties of operations of arrangement of matrices

and multiplication by number:

1. Addition commutative:
.

2. Addition Associative:.

3. There is a zero matrix
satisfying the condition
for all BUT.

4. For any matrix BUTthere is an opposite matrix INsatisfying the condition
.

For any matrices BUTand INand any actual numbers
there are equality:

5.
.

6.
.

7.
.

8.
.

Check property 1. Denote
,
. Let be
,

,
. Have

and since the equality is proved for an arbitrary element, in accordance with the definition of 5
. Property 1 is proved.

Similarly, a property is proved 2.

As a matrix take the order matrix
All the elements of which are zero.

Matching with any matrix according to the rule given in definition 6, we matrix we will not change, and the property is fair.

Check property 4. Let
. Put
. Then
, therefore, property 4 is true.

Check properties 5 - 8 omit.

Definition 8.The work of the matrix
,
,
, on the matrix
,
,
, called the matrix
,
,
, with elements
.

Short record:
.

Example 10. Find the work of Matrix

and
.

In accordance with the definition 8 we will find

Example 11.Multiply matrix

and
.

Note 1. Number of elements in the matrix string equal to the number of elements in the column of the matrix (number of columns of the matrix equal to the number of lines of the matrix ).

Note 2. In the matrix
rows as much as in the matrix and columns as much as in .

Note 3. Generally speaking,
(Multiplication of matrices is noncommutative).

To justify remark 3, it is enough to bring at least one example.

Example 12.Machine in reverse order matrix and from Example 10.

thus, in the general case
.

Note that in a particular case equality
possibly.

Matrians and for which equality is performed
Called permutationor commuting.

Exercises.

1. Find all matrices, permutable with this:

but)
; b)
.

2. Find all second order matrices whose squares are equal to the zero matrix.

3. Prove that
.

Properties of multiplication of matrices:

Multiplication Distribution.

Determinants of the fourth and older orders It is possible to calculate according to the simplified schemes that are to decompose on elements of rows or columns or information to the triangular form. Both methods for clarity will be considered on 4th order matrices.

Method of decomposition on elements of rows or columns

We will consider the first example with detailed explanations of all intermediate actions.

Example 1. Calculate the determinant by decomposition.

Decision. To simplify the calculations, we will define the fourth order determinant for the first line elements (contains a zero element). They are formed by multiplying the items to the appropriate additions to them (the crossing of rows and columns are formed at the intersection of the element for which they are calculated - highlighted in red)

As a result, the calculation will be reduced to the finding of three third-order determinants, which we find on the rule of triangles

Found values \u200b\u200bSubstitute in the output determinant

The result is easy to check with a matrix calculator Yukhymcalc. . To do this, select the matrix-determinant matrix item in the calculator, the matrix size is set to 4 * 4.

The results coincide, therefore, the calculations were carried out correctly.

Example 2. Calculate the fourth-order matrix determinant.

As in the previous task, we will calculate the decomposition method. To do this, choose the elements of the first column. A simplistic determinant can be submitted through the amount of four third-order determinants in the form of

The calculations are not too complicated, the main thing is not imagined with signs and triangles. Found values \u200b\u200bSubstitute to the main determinant and summarize

Equal to the amount of works of elements of some row or column on their algebraic additions, i.e. where i 0 is fixed.
The expression (*) is called the decomposition of the determinant D by line items with the number I 0.

Appointment of service. This service is designed to find the matrix determinant in online mode with the design of the entire progress in Word format. Additionally creates a solution template in Excel.

Instruction. Select the dimension of the matrix, click Next.

Matrix dimension 2 3 4 5 6 7 8 9 10

Calculate the determinant can be in two ways: a-priory and line decomposition or column. If you want to find the determinant to create zeros in one of the rows or columns, you can use this calculator.

Algorithm finding a determinant

1. For matrices of order n \u003d 2, the determinant is calculated by the formula: Δ \u003d A 11 * A 22 -A 12 * A 21
2. For matrices of order n \u003d 3, the determinant is calculated through algebraic additions or by the Sarryus method.
3. The matrix, having a dimension of more than three, is declined to algebraic supplements for which their identifiers (minors) are calculated. For example, 9 order matrix determinant Located through the decomposition of rows or columns (see example).

To calculate the determinant containing the function in the matrix, standard methods apply. For example, calculate the determinant of the matrix of 3 orders:

Use the receipt receipt on the first line.
Δ \u003d sin (x) × + 1 × \u003d 2sin (x) cos (x) -2cos (x) \u003d sin (2x) -2cos (x)

Methods for calculating identifiers

Finding a determinant through algebraic additions It is a common method. Its simplified option is the calculation of the determinant by the Sarryus rule. However, with a large dimension of the matrix, the following methods are used:

1. calculating the determinant by lowering order
2. calculation of the determinant by the Gauss method (through bringing the matrix to triangular form).

In Excel, a function \u003d mopred (cell range) is used to calculate the determinant.

Applied use of determinants

Calculate the determinants, as a rule, for a specific system specified in the form of square matrix. Consider some types of tasks on finding the determinant of the matrix. Sometimes it is required to find an unknown parameter A, in which the determinant would be zero. To do this, it is necessary to make the equation of the determinant (for example, rule of triangles) And, having equated it to 0, calculate the parameter a.
Decomposition of columns (first column):
Minor for (1,1): cross the first string and first column from the matrix.
We find the determinant for this minor. Δ 1.1 \u003d (2 (-2) -2 1) \u003d -6.

We define minor for (2,1): To do this, strike the second line and the first column from the matrix.

We find the determinant for this minor. Δ 2.1 \u003d (0 (-2) -2 (-2)) \u003d 4. Minor for (3,1): strike out the 3rd line and 1st column from the matrix.
We find the determinant for this minor. Δ 3.1 \u003d (0 1-2 (-2)) \u003d 4
The main determinant is: Δ \u003d (1 (-6) -3 4 + 1 4) \u003d -14

We will find the determinant using the strings decomposition (on the first line):
Minor for (1,1): cross the first string and first column from the matrix.

We find the determinant for this minor. Δ 1.1 \u003d (2 (-2) -2 1) \u003d -6. Minor for (1,2): cross the 1st line and 2nd column from the matrix. Calculate the determinant for this minor. Δ 1.2 \u003d (3 (-2) -1 1) \u003d -7. And to find the minor for (1.3), strike out the first string and the third column from the matrix. We find the determinant for this minor. Δ 1,3 \u003d (3 2-1 2) \u003d 4
We find the main determinant: Δ \u003d (1 (-6) -0 (-7) + (- 2 4)) \u003d -14

During the solution of tasks on higher mathematics, it is very often necessary calculate the determinant of the matrix. The determinant of the matrix appears in a linear algebra, analytical geometry, mathematical analysis and other sections of higher mathematics. Thus, without the skill of solutions, the determinants simply could not do. Also for self-test you can download the calculator of the determinants for free, he will not teach the determinants in itself, but it is very convenient because it is always advantageous to know the correct answer in advance!

I will not give a strict mathematical definition of the determinant, and, in general, I will try to minimize mathematical terminology, most of the readers will not be easier. The task of this article is to teach you to solve the identifiers of the second, third and fourth order. All material is set forth in a simple and affordable form, and even the full (empty) kettle in the highest mathematics after a careful study of the material will be able to correctly solve the determinants.

In practice, it is most often possible to meet the determinant of the second order, for example: and the third-order determinant, for example: .

The determinant of the fourth order Also not antiques, and we will come up at the end of the lesson.

I hope everyone understands the following: The numbers inside the determinant live by themselves, and not about what subtraction of speech does not go! You can not change the numbers!

(As speciality, you can carry out paired permutations of rows or columns of the determinant with a change of its sign, but often there is no need for this - see the following lesson property of the determinant and lowering its order)

Thus, if any determinant is given, then nothing inside him do not touch!

Designations: If the matrix is \u200b\u200bgiven The determinant is designated. Also, very often the determinant is indicated by the Latin letter or Greek.

1) What does it mean to decide (find, reveal) the determinant? Calculate the determinant - it means to find the number. The signs of the question in the above examples are quite ordinary numbers.

2) now it remains to figure out How to find this number? To do this, you need to apply certain rules, formulas and algorithms, which now will be discussed.

Let's start with the "two" determinant on "two":

This must be remembered, at least at the time of studying the highest mathematics in the university.

Immediately consider an example:

Ready. The most important thing is not to get confused.

The determinant of the "three three" matrix You can reveal 8 methods, 2 of them are simple and 6 - normal.

Let's start with two simple ways

Similarly, the determinant "two two", the determinant "three to three" can be revealed using the formula:

The formula is long and make an easy error easier than simple. How to avoid annoying misses? For this purpose, a second way to calculate the determinant, which actually coincides with the first one. It is called the Sarruska method or the way of "parallel strips".
The essence is that the first and second column is attributed to the right and the second column and neatly the pencil is carried out:

Farmers located on the "red" diagonals are included in the formula with a plus sign.
The factors located on the "blue" diagonals are included in the formula with a minus sign:

Example:

Compare two solutions. It is easy to notice that this is the same, simply in the second case there are a bit of the factors of the formula, and, most importantly, the probability of allowing the error is significantly less.

Now consider six normal ways to calculate the determinant.

Why normal? Because in the overwhelming majority of cases, the determinants need to be disclosed exactly as.

As you noticed, the "three-three" determinant three columns and three lines.
Solve the determinant can be discontinued on any row or on any column.
Thus, it turns out 6 ways, and in all cases used simplicity algorithm.

The determinant of the matrix is \u200b\u200bequal to the amount of product elements (column) to the corresponding algebraic additions. Scary? Everything is much easier, we will use an unscientific, but understandable approach, affordable even for a person far from mathematics.

In the following example, we will disclose the determinant on the first line.
For this we will need a matrix of signs :. It is easy to notice that signs are located in a checker order.

Attention! Matrix of signs is my own invention. This concept is not scientific, it does not need to be used in the piston design tasks, it only helps you understand the algorithm for calculating the determinant.

First I will give a complete solution. We again take our experimental determinant and carry out computation:

AND main question: How from the "Three Three" determines this is this:
?

So, the determinant "three to three" is reduced to solving three small determinants, or as they are also called Minorov. The term recommended remember, all the more, it is memorable: Minor is small.

Kohl soon chosen the method of decomposition of the determinant on the first lineObviously, everything revolves around it:

Elements are usually considered from left to right (or from top to bottom if a column would be selected)

We went, first we understand with the first element of the string, that is, with a unit:

1) From the matrix of signs, we write the corresponding sign:

2) Then write the element itself:

3) mentally delete the string and column in which the first element is worth:

The remaining four numbers and form the determinant "two two", which is called Minor This element (units).

Go to the second line item.

4) From the matrix of signs we write the corresponding sign:

5) Then write the second element:

6) Mentally delete the string and column in which the second element is worth:

Well, the third element of the first line. No originality:

7) From the matrix of signs we write out the corresponding sign:

8) write the third element:

9) Mentally delete the string and column in which the third element is worth:

The remaining four numbers are recorded in a small determinant.

The remaining actions do not represent difficulties, since the determinants are "two to two" we can already be considered. Do not confuse in signs!

Similarly, the determinant can be decomposed on any row or on any column. Naturally, in all six cases, the answer is equal.

The "Four Four" determines can be calculated using the same algorithm.
At the same time, the matrix of signs will increase:

In the following example, I revealed the determinant on the fourth column:

And how it happened, try to sort it yourself. Additional information will be later. If someone wants to break the determinant to the end, the correct answer: 18. For training it is better to reveal the determinant for some other column or another line.

It is very good and useful to practice, reveal. But how long will you spend on a large determinant? Is it possible somehow faster and more reliable? I propose to get acquainted with effective methods Calculations of determinants in the second lesson - the properties of the determinant. Reducing the order of the determinant.

BE CAREFUL!

Formulation of the problem

The task implies the user's acquaintance with basic concepts numerical methods, such as the determinant and the reverse matrix, and different ways their calculations. In this theoretical report, a simple and affordable language, basic concepts and definitions are first introduced, 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 easily can take advantage of the results of this work. For clarity, a program for calculating the matrix determinant by several methods written in the C ++ programming language. The program is used as a laboratory booth to create illustrations to the report. A study of methods for solving systems of linear algebraic equations is carried out. The uselessness of the calculation of the reverse matrix is \u200b\u200bproved, so the paper provides more optimal ways to solve equations without calculating it. Tells why there is such a number various methods Calculations of determinants and inverse matrices and disassemble their disadvantages. The errors are also considered when calculating the determinant and the accuracy achieved is estimated. In addition to Russian terms, their English equivalents are used in the work, under what names to seek numerical procedures in libraries and what the parameters mean.

Main definitions and simplest properties

Determinant

We introduce the definition of the determinant of the square matrix of any order. This definition will be recurrent, that is, to establish what is the determinant of the order matrix, you need to know what is the determinant of the order matrix. We also note that the determinant exists only in square matrices.

The determinant of the square matrix will be denoted or Det.

Definition 1. Determinant Square matrix second order called the number .

Determinant a square matrix of order, called the number

where - the determinant of the order matrix obtained from the matrix by crossing the first line and the column with the number.

For clarity, write down how can I calculate the fourth order matrix determinant:

Comment. The real calculation of determinants for matrices above the third order based on the definition is used in exceptional cases. As a rule, the calculation is conducted according to other algorithms, which will be discussed later and which require less computational work.

Comment. In definition 1, it would rather say that the determinant is a function defined on the set of square matrices of order and the receiving values \u200b\u200bin a set of numbers.

Comment. Instead of the term "determinant", the term "determinant", which has the same meaning, is also used in the literature. From the word "determinant" and the designation of DET appeared.

Consider some properties of determinants that formulate in the form of statements.

Approval 1. When transposed by the matrix, the determinant does not change, that is.

Approval 2. The determinant of the product of square matrices is equal to the product of the determinants of the factors, that is.

Approval 3. If you change two lines in the matrix, then its determinant will change the sign.

Approval 4. If the matrix has two identical lines, then its determinant is zero.

In the future, we will need to fold the strings and multiply the row by the number. These actions on strings (columns) we will perform the same way as actions on matrices-lines (column matrices), that is, elementary. The result will serve as a string (column), as a rule, which does not match the rows of the original matrix. In the presence of operations of folding strings (columns) and multiplying them, we can talk about linear combinations of strings (columns), that is, amounts with numeric coefficients.

Approval 5. If the matrix string is multiplied by the number, then its determinant multiplies this number.

Approval 6. If the matrix contains a zero string, then its determinant is zero.

Approval 7. If one of the matrix rows is another multiplied by the number (rows are proportional to), then the matrix determinant is zero.

Approval 8. Suppose in the matrix, the I-Aya line has the form. Then, where the matrix is \u200b\u200bobtained from the matrix by replacing the i-th row on the string, and the matrix is \u200b\u200ba replacement of the i-th row to the string.

Approval 9. If you add another multiplied by the matrix to one of the matrix rows, then the matrix determinant will not change.

Approval 10. If one of the matrix lines is a linear combination of its other lines, the matrix determinant is zero.

Definition 2. Algebraic supplement A number equal to the matrix element is called, where is the matrix determinant obtained from the matrix by crossing the I-th row and the J-th column. An algebraic addition to the matrix element is indicated.

Example. Let be . Then

Comment. Using algebraic additions, determining 1 determinant can be written as follows:

Approval 11. Decomposition of the determinant for an arbitrary string.

For determinant, the matrix is \u200b\u200bfair formula

Example. Calculate .

Decision. We use the decomposition of the third line, so more profitable, since in the third line two numbers of three - zeros. Receive

Approval 12. For a square matrix of order when the ratio is performed .

Approval 13. All properties of the determinant, formulated for strings (approval 1 - 11), are valid for columns, in particular, the decomposition of the determinant on the J-th column and equality at.

Approval 14. The determinant of the triangular matrix is \u200b\u200bequal to the product of its main diagonal elements.

Corollary. The determinant of a single matrix equal to unity, .

Output. The above properties make it possible to find determinants of matrices of sufficiently high orders with a relatively small amount of calculations. The calculation algorithm is next.

Algorithm for creating zeros in the column. Let us calculate the order determinant. If, then change the first string and any other 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 zero, the matrix has a zero column and according to statements 1, 13, its determinant is zero.

So, we believe that already in the initial matrix. Leave the first line unchanged. We add the first string to the first line multiplied by the number. Then the first element of the second line will be equal .

The remaining elements of the new second line will be denoted. The determinant of the new matrix according to the statement 9 is equal. The first line will be multiplied by the number and add to the third. The first element of the new third line will be equal

The remaining elements of the new third line will be denoted. The determinant of the new matrix according to the statement 9 is equal.

The process of obtaining zeros instead of the first elements of strings will continue further. Finally, the first line will be multiplied by the number and add to the last line. As a result, the matrix is \u200b\u200bobtained, we denote it, which has the form

and. To calculate the determinant of the matrix, we use decomposition by the first column

Since, then

The definition of the order matrix is \u200b\u200blocated on the right side. It is applicable to it the same algorithm, and the calculation of the determinant of the matrix will be reduced to the calculation of the determinant of the order matrix. The process repeat until we do before the second order determined, which is calculated by definition.

If the matrix does not have any specific properties, then noticeably reduce the amount of calculations compared to the proposed algorithm fails. Another good side of this algorithm - it is easy to make a program for a computer to calculate the determinants of large order matrices. In standard software calculation programs, this algorithm is used with not fundamental changes associated with minimizing the effect of rounding errors and input errors when computer calculations.

Example. Calculate the determinant of the matrix .

Decision. Leave the first string without a change. The second line add the first multiplied by the number:

The determinant does not change. To the third line add the first multiplied by the number:

The determinant does not change. To the fourth line add the first multiplied by the number:

The determinant does not change. As a result, we get

At the same algorithm, we consider the determinant of the matrix of about 3, standing on the right. Leave the first line without changes, adding the first to the second line multiplied by the number :

To the third line add the first multiplied by the number :

As a result, we get

Answer. .

Comment. Although the fractions were used in calculations, the result turned out to be an integer. Indeed, using the properties of the determinants and the fact that the initial numbers are integer, the operations with fractions could be avoided. But in engineering practice, the number is extremely rarely integer. Therefore, as a rule, the elements of the determinant will be decimal fractions and apply some tricks to simplify calculations inexpediently.

inverse matrix

Definition 3. The matrix is \u200b\u200bcalled inverse matrix For a square matrix, if.

From the definition, it follows that the reverse matrix will be a square matrix of the same order as the matrix (otherwise one of the works or it would not be determined).

The inverse matrix for the matrix is \u200b\u200bindicated. Thus, if there is, then.

From the determination of the reverse matrix it follows that the matrix is \u200b\u200breverse for the matrix, that is. About the matrix and you can say that they are reversed each other or mutually reverse.

If the determinant of the matrix is \u200b\u200bzero, then the reverse to it does not exist.

Since it is important for finding the reverse matrix, whether the identifier of the marita is zero or not, then we introduce the following definitions.

Definition 4. Square matrix name degenerate or special matrix, if non-degenerate or non-singular matrix, if a .

Statement. If the reverse matrix exists, then it is unique.

Statement. If the square matrix is \u200b\u200bnon-degenerate, then the reverse for it exists and (1) Where - algebraic additions to elements.

Theorem. The inverse matrix for the square matrix exists if and only if the matrix is \u200b\u200bnondegenerate, the inverse matrix is \u200b\u200bunique, and the formula (1) is valid.

Comment. Special attention should be paid to places occupied by algebraic additions in the return matrix formula: the first index shows the number column, and the second is the number stringsIn which the calculated algebraic addition should be recorded.

Example. .

Decision. We find the determinant

Since, the matrix is \u200b\u200bnondegenerate, and the reverse for it exists. We find algebraic additions:

We compose a reverse matrix, placing the algebraic supplements found so that the first index corresponds to the column, and the second line: (2)

The resulting matrix (2) and serves as a response to the task.

Comment. In the previous example it would be more accurate to record the answer:
(3)

However, recording (2) is more compact and it is more convenient to carry out further computing, if necessary. Therefore, the response entry in the form (2) is preferable if the elements of the matrices are integers. And vice versa, if the elements of the matrix - decimal fractions, the reverse matrix is \u200b\u200bbetter to record without a multiplier ahead.

Comment. When finding a reverse matrix, it is necessary to perform quite a few computing and unusually the rule of algebraic additions in the final matrix. Therefore, the likelihood of error is great. To avoid errors, you should check: calculate the product of the original matrix to the final in one way or another. If the result is a single matrix, the reverse matrix is \u200b\u200bfound correctly. Otherwise, you need to look for an error.

Example. Find the recovery matrix for the matrix .

Decision. - exists.

Answer: .

Output. Finding the reverse matrix by formula (1) requires too many calculations. For fourth-order matrices and above it is unacceptable. The real algorithm for finding a reverse matrix will be given later.

Calculation of the determinant and reverse matrix using the Gauss method

The Gauss method can be used to find the determinant and the return matrix.

It is that the determinant of the matrix is \u200b\u200bDet.

Reverse matrix is \u200b\u200bsolving systems linear equations The exclusion method of Gauss:

Where there is a j-column of a single matrix, is the desired vector.

The resulting vector solutions - form, obviously, the columns of the matrix, since.

Formulas for the determinant

1. If the matrix is \u200b\u200bnon-degenerate, then and (the product of the leading elements).