In this article we will talk about the matrix method for solving a system of linear algebraic equations, find its definition and give examples of solutions.
Definition 1
Inverse matrix method is a method used to solve SLAEs if the number of unknowns is equal to the number of equations.
Example 1
Find a solution to a system of n linear equations with n unknowns:
a 11 x 1 + a 12 x 2 + . . . + a 1 n x n = b 1 a n 1 x 1 + a n 2 x 2 + . . . + a n n x n = b n
Matrix recording type : A × X = B
where A = a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋯ ⋯ ⋯ ⋯ a n 1 a n 2 ⋯ a n n is the matrix of the system.
X = x 1 x 2 ⋮ x n - column of unknowns,
B = b 1 b 2 ⋮ b n - column of free coefficients.
From the equation we received, it is necessary to express X. To do this, you need to multiply both sides of the matrix equation on the left by A - 1:
A - 1 × A × X = A - 1 × B.
Since A - 1 × A = E, then E × X = A - 1 × B or X = A - 1 × B.
Comment
The inverse matrix to matrix A has the right to exist only if the condition d e t A is not equal to zero is satisfied. Therefore, when solving SLAEs using the inverse matrix method, first of all, d e t A is found.
In the event that d e t A is not equal to zero, the system has only one solution option: using the inverse matrix method. If d e t A = 0, then the system cannot be solved by this method.
An example of solving a system of linear equations using the inverse matrix method
Example 2We solve the SLAE using the inverse matrix method:
2 x 1 - 4 x 2 + 3 x 3 = 1 x 1 - 2 x 2 + 4 x 3 = 3 3 x 1 - x 2 + 5 x 3 = 2
How to solve?
- We write the system in the form of a matrix equation A X = B, where
A = 2 - 4 3 1 - 2 4 3 - 1 5, X = x 1 x 2 x 3, B = 1 3 2.
- We express X from this equation:
- Find the determinant of matrix A:
d e t A = 2 - 4 3 1 - 2 4 3 - 1 5 = 2 × (- 2) × 5 + 3 × (- 4) × 4 + 3 × (- 1) × 1 - 3 × (- 2) × 3 - - 1 × (- 4) × 5 - 2 × 4 - (- 1) = - 20 - 48 - 3 + 18 + 20 + 8 = - 25
d e t A does not equal 0, therefore, the inverse matrix solution method is suitable for this system.
- We find the inverse matrix A - 1 using the allied matrix. We calculate the algebraic complements A i j to the corresponding elements of the matrix A:
A 11 = (- 1) (1 + 1) - 2 4 - 1 5 = - 10 + 4 = - 6,
A 12 = (- 1) 1 + 2 1 4 3 5 = - (5 - 12) = 7,
A 13 = (- 1) 1 + 3 1 - 2 3 - 1 = - 1 + 6 = 5,
A 21 = (- 1) 2 + 1 - 4 3 - 1 5 = - (- 20 + 3) = 17,
A 22 = (- 1) 2 + 2 2 3 3 5 - 10 - 9 = 1,
A 23 = (- 1) 2 + 3 2 - 4 3 - 1 = - (- 2 + 12) = - 10,
A 31 = (- 1) 3 + 1 - 4 3 - 2 4 = - 16 + 6 = - 10,
A 32 = (- 1) 3 + 2 2 3 1 4 = - (8 - 3) = - 5,
A 33 = (- 1) 3 + 3 2 - 4 1 - 2 = - 4 + 4 = 0.
- We write down the allied matrix A *, which is composed of algebraic complements of the matrix A:
A * = - 6 7 5 17 1 - 10 - 10 - 5 0
- We write the inverse matrix according to the formula:
A - 1 = 1 d e t A (A *) T: A - 1 = - 1 25 - 6 17 - 10 7 1 - 5 5 - 10 0 ,
- We multiply the inverse matrix A - 1 by the column of free terms B and obtain a solution to the system:
X = A - 1 × B = - 1 25 - 6 17 - 10 7 1 - 5 5 - 10 0 1 3 2 = - 1 25 - 6 + 51 - 20 7 + 3 - 10 5 - 30 + 0 = - 1 0 1
Answer : x 1 = - 1 ; x 2 = 0 ; x 3 = 1
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For any non-singular matrix A there is a unique matrix A -1 such that
A*A -1 =A -1 *A = E,
where E is the identity matrix of the same orders as A. The matrix A -1 is called the inverse of matrix A.
In case someone forgot, in the identity matrix, except for the diagonal filled with ones, all other positions are filled with zeros, an example of an identity matrix:
Finding the inverse matrix using the adjoint matrix method
The inverse matrix is defined by the formula:
where A ij - elements a ij.
Those. To calculate the inverse matrix, you need to calculate the determinant of this matrix. Then find the algebraic complements for all its elements and compose a new matrix from them. Next you need to transport this matrix. And divide each element of the new matrix by the determinant of the original matrix.
Let's look at a few examples.
Find A -1 for a matrix
Solution. Let's find A -1 using the adjoint matrix method. We have det A = 2. Let us find the algebraic complements of the elements of matrix A. In this case, the algebraic complements of the matrix elements will be the corresponding elements of the matrix itself, taken with a sign in accordance with the formula
We have A 11 = 3, A 12 = -4, A 21 = -1, A 22 = 2. We form the adjoint matrix
We transport the matrix A*:
We find the inverse matrix using the formula:
We get:
Using the adjoint matrix method, find A -1 if
Solution. First of all, we calculate the definition of this matrix to verify the existence of the inverse matrix. We have
Here we added to the elements of the second row the elements of the third row, previously multiplied by (-1), and then expanded the determinant for the second row. Since the definition of this matrix is different from zero, then its inverse matrix exists. To construct the adjoint matrix, we find the algebraic complements of the elements of this matrix. We have
According to the formula
transport matrix A*:
Then according to the formula
Finding the inverse matrix using the method of elementary transformations
In addition to the method of finding the inverse matrix, which follows from the formula (the adjoint matrix method), there is a method for finding the inverse matrix, called the method of elementary transformations.
Elementary matrix transformations
The following transformations are called elementary matrix transformations:
1) rearrangement of rows (columns);
2) multiplying a row (column) by a number other than zero;
3) adding to the elements of a row (column) the corresponding elements of another row (column), previously multiplied by a certain number.
To find the matrix A -1, we construct a rectangular matrix B = (A|E) of orders (n; 2n), assigning to matrix A on the right the identity matrix E through a dividing line:
Let's look at an example.
Using the method of elementary transformations, find A -1 if
Solution. We form matrix B:
Let us denote the rows of matrix B by α 1, α 2, α 3. Let us perform the following transformations on the rows of matrix B.
The inverse matrix for a given matrix is such a matrix, multiplying the original one by which gives the identity matrix: A mandatory and sufficient condition for the presence of an inverse matrix is that the determinant of the original one is not equal to zero (which in turn implies that the matrix must be square). If the determinant of a matrix is equal to zero, then it is called singular and such a matrix does not have an inverse. In higher mathematics, inverse matrices are important and are used to solve a number of problems. For example, on finding the inverse matrix a matrix method for solving systems of equations was constructed. Our service site allows calculate inverse matrix online two methods: the Gauss-Jordan method and using the matrix of algebraic additions. The first one involves a large number of elementary transformations inside the matrix, the second one involves the calculation of the determinant and algebraic additions to all elements. To calculate the determinant of a matrix online, you can use our other service - Calculation of the determinant of a matrix online
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Let there be a square matrix of nth order
Matrix A -1 is called inverse matrix in relation to 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 the 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 in which the number of rows and columns coincide.
Theorem for the existence condition of an inverse matrix
In order for a matrix to have an inverse matrix, it is necessary and sufficient that it be non-singular.
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 matrix A into the table for solving systems of equations using the Gaussian method and assign matrix E to it on the right (in place of the right-hand sides of the equations).
- Using Jordan transformations, reduce matrix A to a matrix consisting of unit 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 under the matrix A of the original table you get the identity matrix E.
- Write down the inverse matrix A -1, which is located in the last table under the matrix E of the original table.
For matrix A, find the inverse matrix A -1
Solution: We write matrix A and assign the identity matrix E to the right. Using Jordan transformations, we reduce matrix A to the identity matrix E. The calculations are given 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 was obtained. Therefore, the calculations were made correctly.
Answer: 
Solving matrix equations
Matrix equations can look like:
AX = B, HA = B, AXB = C,
where A, B, C are the specified matrices, X is the desired matrix.
Matrix equations are solved by multiplying the equation by inverse matrices.
For example, to find the matrix from the 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 matrix is equal to (see example 1)
Matrix method in economic analysis
Along with others, they are also used 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 make a comparative assessment of the functioning of organizations and their structural divisions.
In the process of applying matrix analysis methods, several stages can be distinguished.
At the first stage a system of economic indicators is being formed and on its basis a matrix of initial data is compiled, which is a table in which system numbers are shown in its individual rows (i = 1,2,....,n), and in vertical columns - numbers of indicators (j = 1,2,....,m).
At the second stage For each vertical column, the largest of the available indicator values is identified, which is taken as one.
After this, all 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 matrix indicator is assigned a certain weight coefficient k. The value of the latter is determined by expert opinion.
On the last one, fourth stage found rating values Rj are grouped in order of their increase or decrease.
The matrix methods outlined should be used, for example, in a comparative analysis of various investment projects, as well as in assessing other economic indicators of the activities of organizations.
Let's continue the conversation about actions with matrices. Namely, during the study of this lecture you will learn how to find the inverse matrix. Learn. Even if math is difficult.
What is an inverse matrix? Here we can draw an analogy with inverse numbers: consider, for example, the optimistic number 5 and its inverse number. The product of these numbers is equal to one: . Everything is similar with matrices! The product of a matrix and its inverse matrix is equal to – identity matrix, which is the matrix analogue of the numerical unit. However, first things first – let’s first solve an important practical issue, namely, learn how to find this very inverse matrix.
What do you need to know and be able to do to find the inverse matrix? You must be able to decide qualifiers. You must understand what it is matrix and be able to perform some actions with them.
There are two main methods for finding the inverse matrix:
by using algebraic additions And using elementary transformations.
Today we will study the first, simpler method.
Let's start with the most terrible and incomprehensible. Let's consider square matrix. The inverse matrix can be found using the following formula:
Where is the determinant of the matrix, is the transposed matrix of algebraic complements of the corresponding elements of the matrix.
The concept of an inverse matrix exists only for square matrices, matrices “two by two”, “three by three”, etc.
Designations: As you may have already noticed, the inverse matrix is denoted by a superscript
Let's start with the simplest case - a two-by-two matrix. Most often, of course, “three by three” is required, but, nevertheless, I strongly recommend studying a simpler task in order to understand the general principle of the solution.
Example:
Find the inverse of a matrix
Let's decide. It is convenient to break down the sequence of actions point by point.
1) First we find the determinant of the matrix.
If your understanding of this action is not good, read the material How to calculate the determinant?
Important! If the determinant of the matrix is equal to ZERO– inverse matrix DOESN'T EXIST.
In the example under consideration, as it turned out, , which means everything is in order.
2) Find the matrix of minors.
To solve our problem, it is not necessary to know what a minor is, however, it is advisable to read the article How to calculate the determinant.
The matrix of minors has the same dimensions as the matrix, that is, in this case.
The only thing left to do is find four numbers and put them instead of asterisks.
Let's return to our matrix
Let's look at the top left element first:
How to find it minor?
And this is done like this: MENTALLY cross out the row and column in which this element is located:
The remaining number is minor of this element, which we write in our matrix of minors:
Consider the following matrix element:
Mentally cross out the row and column in which this element appears:
What remains is the minor of this element, which we write in our matrix:
Similarly, we consider the elements of the second row and find their minors:
Ready.
It's simple. In the matrix of minors you need CHANGE SIGNS two numbers:
These are the numbers that I circled!
– matrix of algebraic additions of the corresponding elements of the matrix.
And just...
4) Find the transposed matrix of algebraic additions.
– transposed matrix of algebraic complements of the corresponding elements of the matrix.
5) Answer.
Let's remember our formula
Everything has been found!
So the inverse matrix is: 
It is better to leave the answer as is. NO NEED divide each element of the matrix by 2, since the result is fractional numbers. This nuance is discussed in more detail in the same article. Actions with matrices.
How to check the solution?
You need to perform matrix multiplication or
Examination: 
Received already mentioned identity matrix is a matrix with ones by main diagonal and zeros in other places.
Thus, the inverse matrix is found correctly.
If you carry out the action, the result will also be an identity matrix. This is one of the few cases where matrix multiplication is commutative, more details can be found in the article Properties of operations on matrices. Matrix Expressions. Also note that during the check, the constant (fraction) is brought forward and processed at the very end - after the matrix multiplication. This is a standard technique.
Let's move on to a more common case in practice - the three-by-three matrix:
Example:
Find the inverse of a matrix 
The algorithm is exactly the same as for the “two by two” case.
We find the inverse matrix using the formula: , where is the transposed matrix of algebraic complements of the corresponding elements of the matrix.
1) Find the determinant of the matrix.
Here the determinant is revealed on the first line.
Also, don’t forget that, which means everything is fine - inverse matrix exists.
2) Find the matrix of minors.
The matrix of minors has a dimension of “three by three” 
, and we need to find nine numbers.
I'll take a closer look at a couple of minors:
Consider the following matrix element: 
MENTALLY cross out the row and column in which this element is located: 
We write the remaining four numbers in the “two by two” determinant. 
This two-by-two determinant and is the minor of this element. It needs to be calculated: 
That’s it, the minor has been found, we write it in our matrix of minors: 
As you probably guessed, you need to calculate nine two-by-two determinants. The process, of course, is tedious, but the case is not the most severe, it can be worse.
Well, to consolidate – finding another minor in the pictures: 
Try to calculate the remaining minors yourself.
Final result: 
– matrix of minors of the corresponding elements of the matrix.
The fact that all the minors turned out to be negative is purely an accident.
3) Find the matrix of algebraic additions.
In the matrix of minors it is necessary CHANGE SIGNS strictly for the following elements: 
In this case:
We do not consider finding the inverse matrix for a “four by four” matrix, since such a task can only be given by a sadistic teacher (for the student to calculate one “four by four” determinant and 16 “three by three” determinants). In my practice, there was only one such case, and the customer of the test paid quite dearly for my torment =).
In a number of textbooks and manuals you can find a slightly different approach to finding the inverse matrix, but I recommend using the solution algorithm outlined above. Why? Because the likelihood of getting confused in calculations and signs is much less.
