Rules for solving matrices. Solve the matrix problem yourself, and then see the solution. Matrix eigenvalues and matrix eigenvectors
Matrix A -1 is called the inverse matrix with respect to matrix A, if A * A -1 \u003d E, where E is the identity matrix of the nth order. The inverse matrix can only exist for square matrices.
Service assignment. Using this service online, you can find algebraic additions, transposed matrix A T , union matrix and inverse matrix. The solution is carried out directly on the site (online) and is free. The calculation results are presented in a report in Word format and in Excel format (that is, it is possible to check the solution). see design example.
Instruction. To obtain a solution, you must specify the dimension of the matrix. Next, in the new dialog box, fill in the matrix A .
See also Inverse Matrix by the Jordan-Gauss Method
Algorithm for finding the inverse matrix
- Finding the transposed matrix A T .
- Definition of algebraic additions. Replace each element of the matrix with its algebraic complement.
- Compilation of an inverse matrix from algebraic additions: each element of the resulting matrix is divided by the determinant of the original matrix. The resulting matrix is the inverse of the original matrix.
- Determine if the matrix is square. If not, then there is no inverse matrix for it.
- Calculation of the determinant of the matrix A . If he doesn't zero, we continue the solution, otherwise - the inverse matrix does not exist.
- Definition of algebraic additions.
- Filling in the union (mutual, adjoint) matrix C .
- Compilation of the inverse matrix from algebraic additions: each element of the adjoint matrix C is divided by the determinant of the original matrix. The resulting matrix is the inverse of the original matrix.
- Make a check: multiply the original and the resulting matrices. The result should be an identity matrix.
Example #1. We write the matrix in the form:
Algebraic additions.
A 1.1 = (-1) 1+1 |
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∆ 1,1 = (-1 4-5 (-2)) = 6
A 1,2 = (-1) 1+2 |
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∆ 1,2 = -(2 4-(-2 (-2))) = -4
A 1.3 = (-1) 1+3 |
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∆ 1,3 = (2 5-(-2 (-1))) = 8
A 2.1 = (-1) 2+1 |
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∆ 2,1 = -(2 4-5 3) = 7
A 2.2 = (-1) 2+2 |
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∆ 2,2 = (-1 4-(-2 3)) = 2
A 2.3 = (-1) 2+3 |
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∆ 2,3 = -(-1 5-(-2 2)) = 1
A 3.1 = (-1) 3+1 |
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∆ 3,1 = (2 (-2)-(-1 3)) = -1
A 3.2 = (-1) 3+2 |
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∆ 3,2 = -(-1 (-2)-2 3) = 4
A 3.3 = (-1) 3+3 |
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∆ 3,3 = (-1 (-1)-2 2) = -3
Then inverse matrix can be written as:
A -1 = 1 / 10 |
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A -1 = |
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Another algorithm for finding the inverse matrix
We present another scheme for finding the inverse matrix.- Find the determinant of the given square matrix A .
- We find algebraic additions to all elements of the matrix A .
- We write the algebraic complements of the elements of the rows into the columns (transposition).
- We divide each element of the resulting matrix by the determinant of the matrix A .
A special case: The inverse, with respect to the identity matrix E , is the identity matrix E .
Given Toolkit will help you learn how to matrix operations: addition (subtraction) of matrices, transposition of a matrix, multiplication of matrices, finding the inverse of a matrix. All material is presented in a simple and accessible form, relevant examples are given, so even an unprepared person can learn how to perform actions with matrices. For self-control and self-test, you can download a matrix calculator for free >>>.
I will try to minimize theoretical calculations, in some places explanations “on the fingers” and the use of unscientific terms are possible. Lovers of solid theory, please do not engage in criticism, our task is learn how to work with matrices.
For SUPER-FAST preparation on the topic (who "burns") there is an intensive pdf-course Matrix, determinant and offset!
A matrix is a rectangular table of some elements. As elements we will consider numbers, that is, numerical matrices. ELEMENT is a term. It is desirable to remember the term, it will often occur, it is no coincidence that I used bold to highlight it.
Designation: matrices are usually denoted by capital Latin letters
Example: Consider a two-by-three matrix:
This matrix consists of six elements:
All numbers (elements) inside the matrix exist on their own, that is, there is no question of any subtraction:
It's just a table (set) of numbers!
We will also agree do not rearrange number, unless otherwise stated in the explanation. Each number has its own location, and you cannot shuffle them!
The matrix in question has two rows:
and three columns:
STANDARD: when talking about the dimensions of the matrix, then first indicate the number of rows, and only then - the number of columns. We have just broken down the two-by-three matrix.
If the number of rows and columns of a matrix is the same, then the matrix is called square, for example: is a three-by-three matrix.
If the matrix has one column or one row, then such matrices are also called vectors.
In fact, we know the concept of a matrix since school, consider, for example, a point with coordinates "x" and "y": . Essentially, the coordinates of a point are written into a one-by-two matrix. By the way, here is an example for you why the order of numbers matters: and are two completely different points of the plane.
Now let's move on to the study. matrix operations:
1) Action one. Removing a minus from a matrix (Introducing a minus into a matrix).
Back to our matrix . As you probably noticed, there are too many negative numbers in this matrix. This is very inconvenient in terms of performing various actions with the matrix, it is inconvenient to write so many minuses, and it just looks ugly in the design.
Let's move the minus outside the matrix by changing the sign of EACH element of the matrix:
At zero, as you understand, the sign does not change, zero - it is also zero in Africa.
Reverse example: . Looks ugly.
We introduce a minus into the matrix by changing the sign of EACH element of the matrix:
Well, it's much prettier. And, most importantly, it will be EASIER to perform any actions with the matrix. Because there is such a mathematical folk omen: the more minuses - the more confusion and errors.
2) Action two. Multiplying a Matrix by a Number.
Example:
It's simple, in order to multiply a matrix by a number, you need each multiply the matrix element by the given number. In this case, three.
Another useful example:
– multiplication of a matrix by a fraction
Let's first look at what to do NO NEED:
It is NOT NECESSARY to enter a fraction into the matrix, firstly, it only makes further actions with the matrix difficult, and secondly, it makes it difficult for the teacher to check the solution (especially if - the final answer of the task).
And especially, NO NEED divide each element of the matrix by minus seven:
From the article Mathematics for dummies or where to start, we remember that decimal fractions with a comma in higher mathematics are trying in every possible way to avoid.
The only thing desirable to do in this example is to insert a minus into the matrix:
But if ALL matrix elements were divided by 7 without a trace, then it would be possible (and necessary!) to divide.
Example:
In this case, you can NEED multiply all elements of the matrix by , since all numbers in the matrix are divisible by 2 without a trace.
Note: in theory higher mathematics there is no school concept of "division". Instead of the phrase "this is divided by this", you can always say "this is multiplied by a fraction." That is, division is a special case of multiplication.
3) Action three. Matrix transposition.
To transpose a matrix, you need to write its rows into the columns of the transposed matrix.
Example:
Transpose Matrix
There is only one line here and, according to the rule, it must be written in a column:
is the transposed matrix.
The transposed matrix is usually denoted by a superscript or a stroke on the top right.
Step by step example:
Transpose Matrix
First, we rewrite the first row into the first column:
Then we rewrite the second row into the second column:
And finally, we rewrite the third row into the third column:
Ready. Roughly speaking, to transpose means to turn the matrix on its side.
4) Action four. Sum (difference) of matrices.
The sum of matrices is a simple operation.
NOT ALL MATRIXES CAN BE FOLDED. To perform addition (subtraction) of matrices, it is necessary that they be the SAME SIZE.
For example, if a two-by-two matrix is given, then it can only be added to a two-by-two matrix and no other!
Example:
Add matrices and
To add matrices, you need to add their corresponding elements:
For the difference of matrices, the rule is similar, it is necessary to find the difference of the corresponding elements.
Example:
Find difference of matrices ,
How to decide given example easier to avoid confusion? It is advisable to get rid of unnecessary minuses, for this we will add a minus to the matrix:
Note: in the theory of higher mathematics there is no school concept of "subtraction". Instead of the phrase “subtract this from this”, you can always say “add a negative number to this”. That is, subtraction is a special case of addition.
5) Action five. Matrix multiplication.
What matrices can be multiplied?
For a matrix to be multiplied by a matrix, so that the number of columns of the matrix is equal to the number of rows of the matrix.
Example:
Is it possible to multiply a matrix by a matrix?
So, you can multiply the data of the matrix.
But if the matrices are rearranged, then, in this case, multiplication is no longer possible!
Therefore, multiplication is impossible:
It is not uncommon for tasks with a trick, when a student is asked to multiply matrices, the multiplication of which is obviously impossible.
It should be noted that in some cases it is possible to multiply matrices in both ways.
For example, for matrices, and both multiplication and multiplication are possible
Let there be a square matrix of the nth order
Matrix A -1 is called inverse matrix with respect to the matrix A, if A * A -1 = E, where E is the identity matrix of the nth order.
Identity matrix- such a square matrix, in which all elements along the main diagonal, passing from the upper left corner to the lower right corner, are ones, and the rest are zeros, for example:
inverse matrix may exist only for square matrices those. for those matrices that have the same number of rows and columns.
Inverse Matrix Existence Condition Theorem
For a matrix to have an inverse matrix, it is necessary and sufficient that it be nondegenerate.
The matrix A = (A1, A2,...A n) is called non-degenerate if the column vectors are linearly independent. The number of linearly independent column vectors of a matrix is called the rank of the matrix. Therefore, we can say that in order for an inverse matrix to exist, it is necessary and sufficient that the rank of the matrix is equal to its dimension, i.e. r = n.
Algorithm for finding the inverse matrix
- Write the matrix A in the table for solving systems of equations by the Gauss method and on the right (in place of the right parts of the equations) assign matrix E to it.
- Using Jordan transformations, bring matrix A to a matrix consisting of single columns; in this case, it is necessary to simultaneously transform the matrix E.
- If necessary, rearrange the rows (equations) of the last table so that the identity matrix E is obtained under the matrix A of the original table.
- Write the inverse matrix A -1, which is in the last table under the matrix E of the original table.
For matrix A, find the inverse matrix A -1
Solution: We write down the matrix A and on the right we assign the identity matrix E. Using the Jordan transformations, we reduce the matrix A to the identity matrix E. The calculations are shown in Table 31.1.
Let's check the correctness of the calculations by multiplying the original matrix A and the inverse matrix A -1.
As a result of matrix multiplication, the identity matrix is obtained. Therefore, the calculations are correct.
Answer:
Solution of matrix equations
Matrix equations can look like:
AX = B, XA = B, AXB = C,
where A, B, C are given matrices, X is the desired matrix.
Matrix equations are solved by multiplying the equation by inverse matrices.
For example, to find the matrix from an equation, you need to multiply this equation by on the left.
Therefore, to find a solution to the equation, you need to find the inverse matrix and multiply it by the matrix on the right side of the equation.
Other equations are solved similarly.
Example 2Solve the equation AX = B if
Solution: Since the inverse of the matrix equals (see example 1)
Matrix method in economic analysis
Along with others, they also find application matrix methods. These methods are based on linear and vector-matrix algebra. Such methods are used for the purposes of analyzing complex and multidimensional economic phenomena. Most often, these methods are used when it is necessary to compare the functioning of organizations and their structural divisions.
In the process of applying matrix methods of analysis, several stages can be distinguished.
At the first stage the formation of a system of economic indicators is carried out and on its basis a matrix of initial data is compiled, which is a table in which system numbers are shown in its individual lines (i = 1,2,....,n), and along the vertical graphs - numbers of indicators (j = 1,2,....,m).
At the second stage for each vertical column, the largest of the available values of the indicators is revealed, which is taken as a unit.
After that, all the amounts reflected in this column are divided by highest value and a matrix of standardized coefficients is formed.
At the third stage all components of the matrix are squared. If they have different significance, then each indicator of the matrix is assigned a certain weighting coefficient k. The value of the latter is determined by an expert.
On the last fourth stage found values of ratings Rj grouped in order of increasing or decreasing.
The above matrix methods should be used, for example, when comparative analysis various investment projects, as well as in assessing other economic performance indicators of organizations.
DEFINITION OF A MATRIX. TYPES OF MATRIXES
Matrix size m× n is called the totality m n numbers arranged in a rectangular table of m lines and n columns. This table is usually enclosed in parentheses. For example, the matrix might look like:
For brevity, the matrix can be denoted by a single capital letter, for example, BUT or AT.
In general, a matrix of size m× n write like this
.
The numbers that make up a matrix are called matrix elements. It is convenient to supply matrix elements with two indices aij: The first indicates the row number and the second indicates the column number. For example, a 23– the element is in the 2nd row, 3rd column.
If the number of rows in a matrix is equal to the number of columns, then the matrix is called square, and the number of its rows or columns is called in order matrices. In the examples above, the second matrix is square - its order is 3, and the fourth matrix - its order is 1.
A matrix in which the number of rows is not equal to the number of columns is called rectangular. In the examples, this is the first matrix and the third.
There are also matrices that have only one row or one column.
A matrix with only one row is called matrix - row(or string), and a matrix that has only one column, matrix - column.
A matrix in which all elements are equal to zero is called null and is denoted by (0), or simply 0. For example,
.
main diagonal A square matrix is the diagonal going from the upper left to the lower right corner.
A square matrix in which all elements below the main diagonal are equal to zero is called triangular matrix.
.
A square matrix in which all elements, except perhaps those on the main diagonal, are equal to zero, is called diagonal matrix. For example, or.
A diagonal matrix in which all diagonal entries are equal to one is called single matrix and is denoted by the letter E. For example, the 3rd order identity matrix has the form .
ACTIONS ON MATRIXES
Matrix equality. Two matrices A and B are said to be equal if they have the same number of rows and columns and their corresponding elements are equal aij = b ij. So if and , then A=B, if a 11 = b 11, a 12 = b 12, a 21 = b 21 and a 22 = b 22.
transposition. Consider an arbitrary matrix A from m lines and n columns. It can be associated with the following matrix B from n lines and m columns, where each row is a column of the matrix A with the same number (hence each column is a row of the matrix A with the same number). So if , then .
This matrix B called transposed matrix A, and the transition from A to B transposition.
Thus, transposition is a reversal of the roles of rows and columns of a matrix. Matrix transposed to matrix A, usually denoted A T.
Communication between the matrix A and its transposed can be written as .
For example. Find the matrix transposed to the given one.
Matrix addition. Let matrices A and B consist of the same number of rows and the same number of columns, i.e. have same sizes. Then in order to add the matrices A and B need to matrix elements A add matrix elements B standing in the same places. Thus, the sum of two matrices A and B called matrix C, which is determined by the rule, for example,
Examples. Find the sum of matrices:
It is easy to check that matrix addition obeys the following laws: commutative A+B=B+A and associative ( A+B)+C=A+(B+C).
Multiplying a matrix by a number. To multiply a matrix A per number k need each element of the matrix A multiply by that number. So the matrix product A per number k there is a new matrix, which is determined by the rule or .
For any numbers a and b and matrices A and B equalities are fulfilled:
Examples.
Matrix multiplication. This operation is carried out according to a peculiar law. First of all, we note that the sizes of the matrix factors must be consistent. You can multiply only those matrices whose number of columns of the first matrix matches the number of rows of the second matrix (i.e. the length of the first row is equal to the height of the second column). work matrices A not a matrix B called the new matrix C=AB, whose elements are composed as follows:
Thus, for example, in order to get the product (i.e., in the matrix C) the element in the 1st row and 3rd column from 13, you need to take the 1st row in the 1st matrix, the 3rd column in the 2nd, and then multiply the row elements by the corresponding column elements and add the resulting products. And other elements of the product matrix are obtained using a similar product of the rows of the first matrix by the columns of the second matrix.
In general, if we multiply the matrix A = (aij) size m× n to matrix B = (bij) size n× p, then we get the matrix C size m× p, whose elements are calculated as follows: element c ij is obtained as a result of the product of elements i th row of the matrix A on the relevant elements j-th column of the matrix B and their summation.
From this rule it follows that you can always multiply two square matrices of the same order, as a result we get a square matrix of the same order. In particular, a square matrix can always be multiplied by itself, i.e. square up.
Another important case is the multiplication of a matrix-row by a matrix-column, and the width of the first must be equal to the height of the second, as a result we get a matrix of the first order (i.e. one element). Really,
.
Examples.
Thus, these simple examples show that matrices, generally speaking, do not commute with each other, i.e. A∙B ≠ B∙A . Therefore, when multiplying matrices, you need to carefully monitor the order of the factors.
It can be verified that matrix multiplication obeys the associative and distributive laws, i.e. (AB)C=A(BC) and (A+B)C=AC+BC.
It is also easy to check that when multiplying a square matrix A to the identity matrix E of the same order, we again obtain the matrix A, moreover AE=EA=A.
The following curious fact may be noted. As is known, the product of 2 non-zero numbers is not equal to 0. For matrices, this may not be the case, i.e. the product of 2 non-zero matrices may be equal to the zero matrix.
For example, if , then
.
THE CONCEPT OF DETERMINERS
Let a second-order matrix be given - a square matrix consisting of two rows and two columns .
Second order determinant corresponding to this matrix is the number obtained as follows: a 11 a 22 – a 12 a 21.
The determinant is denoted by the symbol .
So, in order to find the second-order determinant, you need to subtract the product of the elements along the second diagonal from the product of the elements of the main diagonal.
Examples. Calculate second order determinants.
Similarly, we can consider a matrix of the third order and the corresponding determinant.
Third order determinant, corresponding to a given square matrix of the third order, is a number denoted and obtained as follows:
.
Thus, this formula gives the expansion of the third order determinant in terms of the elements of the first row a 11 , a 12 , a 13 and reduces the calculation of the third order determinant to the calculation of second order determinants.
Examples. Calculate the third order determinant.
Similarly, one can introduce the concepts of determinants of the fourth, fifth, etc. orders, lowering their order by expansion over the elements of the 1st row, while the signs "+" and "-" for the terms alternate.
So, unlike the matrix, which is a table of numbers, the determinant is a number that is assigned in a certain way to the matrix.