Check the inverse matrix. Matrix method for solving slough: an example of solving using an inverse matrix. Finding the Inverse Matrix by Gaussian Elimination of Unknowns
Typically, inverse operations are used to simplify complex algebraic expressions. For example, if the problem contains the operation of division by a fraction, you can replace it with the operation of multiplying by a reciprocal, which is the inverse operation. Moreover, matrices cannot be divided, so you need to multiply by the inverse matrix. Calculating the inverse of a 3x3 matrix is quite tedious, but you need to be able to do it manually. You can also find the reciprocal with a good graphing calculator.
Steps
Using the attached matrix
Transpose the original matrix. Transposition is the replacement of rows with columns relative to the main diagonal of the matrix, that is, you need to swap the elements (i, j) and (j, i). In this case, the elements of the main diagonal (starts in the upper left corner and ends in the lower right corner) do not change.
- To swap rows for columns, write the elements of the first row in the first column, the elements of the second row in the second column, and the elements of the third row in the third column. The order of changing the position of the elements is shown in the figure, in which the corresponding elements are circled with colored circles.
Find the definition of each 2x2 matrix. Each element of any matrix, including the transposed one, is associated with a corresponding 2x2 matrix. To find a 2x2 matrix that corresponds to a certain element, cross out the row and column in which this element is located, that is, you need to cross out five elements of the original 3x3 matrix. Four elements that are elements of the corresponding 2x2 matrix will remain uncrossed out.
- For example, to find the 2x2 matrix for the element that is located at the intersection of the second row and the first column, cross out the five elements that are in the second row and first column. The remaining four elements are elements of the corresponding 2x2 matrix.
- Find the determinant of each 2x2 matrix. To do this, subtract the product of the elements of the secondary diagonal from the product of the elements of the main diagonal (see figure).
- Detailed information about 2x2 matrices corresponding to certain elements of a 3x3 matrix can be found on the Internet.
Create a matrix of cofactors. Record the results obtained earlier in the form of a new matrix of cofactors. To do this, write the found determinant of each 2x2 matrix where the corresponding element of the 3x3 matrix was located. For example, if a 2x2 matrix is considered for the element (1,1), write down its determinant in position (1,1). Then change the signs of the corresponding elements according to a certain pattern, which is shown in the figure.
- Sign change scheme: the sign of the first element of the first line does not change; the sign of the second element of the first line is reversed; the sign of the third element of the first line does not change, and so on line by line. Please note that the signs "+" and "-", which are shown in the diagram (see figure), do not indicate that the corresponding element will be positive or negative. In this case, the “+” sign indicates that the sign of the element does not change, and the “-” sign indicates that the sign of the element has changed.
- Detailed information about cofactor matrices can be found on the Internet.
- This is how you find the associated matrix of the original matrix. It is sometimes called the complex conjugate matrix. Such a matrix is denoted as adj(M).
Divide each element of the adjoint matrix by the determinant. The determinant of the matrix M was calculated at the very beginning to check that the inverse matrix exists. Now divide each element of the adjoint matrix by this determinant. Record the result of each division operation where the corresponding element is located. So you will find the matrix, the inverse of the original.
- The determinant of the matrix shown in the figure is 1. Thus, the associated matrix here is the inverse matrix (because dividing any number by 1 does not change it).
- In some sources, the division operation is replaced by the multiplication operation by 1/det(M). In this case, the end result does not change.
Write down the inverse matrix. Write the elements located on the right half of the large matrix as a separate matrix, which is an inverse matrix.
Using a calculator
Choose a calculator that works with matrices. Simple calculators cannot find the inverse matrix, but it can be done with a good graphing calculator such as the Texas Instruments TI-83 or TI-86.
Enter the original matrix into the calculator's memory. To do this, click the Matrix button, if available. For a Texas Instruments calculator, you may need to press the 2 nd and Matrix buttons.
Select the Edit menu. Do this using the arrow buttons or the corresponding function button located at the top of the calculator's keyboard (the location of the button depends on the calculator model).
Enter the matrix designation. Most graphing calculators can work with 3-10 matrices, which can be denoted letters A-J. As a general rule, just select [A] to denote the original matrix. Then press the Enter button.
Enter the matrix size. This article talks about 3x3 matrices. But graphical calculators can work with large matrices. Enter the number of rows, press the Enter button, then enter the number of columns and press the Enter button again.
Enter each element of the matrix. A matrix will be displayed on the calculator screen. If a matrix has already been entered into the calculator before, it will appear on the screen. The cursor will highlight the first element of the matrix. Enter the value of the first element and press Enter. The cursor will automatically move to the next element of the matrix.
Consider a square matrix . Denote by Δ = det A its determinant. A square B is (OM) for a square A of the same order if their product A*B = B*A = E, where E is the identity matrix of the same order as A and B.
A square A is called non-degenerate, or non-singular, if its determinant is non-zero, and degenerate, or special, if Δ = 0.
Theorem. In order for A to have an inverse, it is necessary and sufficient that its determinant be different from zero.
(OM) A, denoted by A -1, so that B \u003d A -1 and is calculated by the formula
, (1)
where А i j - algebraic complements of elements a i j , Δ = detA.
Calculating A -1 by formula (1) for high-order matrices is very laborious, so in practice it is convenient to find A -1 using the method of elementary transformations (EP). Any non-singular A by means of EP of only columns (or only rows) can be reduced to unit E. If EPs performed over the matrix A are applied in the same order to unit E, then the result will be A -1 . It is convenient to perform an EP on A and E at the same time, writing both side by side through the line A|E. If you want to find A -1 , you should use only rows or only columns in your conversions.
Finding the Inverse Matrix Using Algebraic Complements
Example 1. For 
find A -1 .
Solution. We first find the determinant A 
hence, (OM) exists and we can find it by the formula: 
, where A i j (i,j=1,2,3) - algebraic complements of elements a i j of the original A.
The algebraic complement of the element a ij is the determinant or minor M ij . It is obtained by deleting column i and row j. The minor is then multiplied by (-1) i+j , i.e. A ij =(-1) i+j M ij
where 
.
Finding the inverse matrix using elementary transformations
Example 2. Using the method of elementary transformations, find A -1 for: A \u003d.
Solution. We attribute to the original A on the right a unit of the same order: 
. With the help of elementary column transformations, we bring the left “half” to the unit one, simultaneously performing exactly such transformations on the right “half”.
To do this, swap the first and second columns: 
~
. We add the first to the third column, and the first multiplied by -2 to the second: 
. From the first column we subtract the doubled second, and from the third - the second multiplied by 6; 
. Let's add the third column to the first and second: 
. Multiply the last column by -1: 
. The square table obtained to the right of the vertical bar is the inverse of A -1. So, 
.
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 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.
Solve 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 the largest 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, in a comparative analysis of various investment projects, as well as in assessing other economic performance indicators of organizations.
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.
- Composing 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 it is not equal to 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:
| A -1 = |
|
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 .