Find the inverse matrix solution. Inverse matrix and its properties
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 matrix 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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website allows you to find inverse matrix online fast and free. On the site, calculations are made using our service and the result is given with a detailed solution for finding inverse matrix. The server always gives only an accurate and correct answer. In tasks by definition inverse matrix online, it is necessary that the determinant matrices was nonzero, otherwise website will report the impossibility of finding the inverse matrix due to the fact that the determinant of the original matrix is equal to zero. The task of finding inverse matrix found in many branches of mathematics, being one of the most basic concepts of algebra and a mathematical tool in applied problems. Independent definition of inverse matrix requires significant effort, a lot of time, calculations and great care to avoid typos or minor errors in calculations. Therefore our service finding the inverse matrix online will make your task much easier and will become an indispensable tool for solving mathematical problems. Even if you find the inverse matrix yourself, we recommend checking your solution on our server. Enter your original matrix at our Calculation of inverse matrix online and check your answer. Our system never makes mistakes and finds inverse matrix given dimension in mode online instantly! On the site website character entries are allowed in elements matrices, in this case inverse matrix online will be presented in general symbolic form.
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 nonzero, 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.
Matrix A -1 is called the inverse matrix with respect to matrix A if A*A -1 = E, where E is the identity matrix of the nth order. An inverse matrix can only exist for square matrices.
Purpose of the service. Using this service online you can find algebraic complements, transposed matrix A T, allied matrix and inverse matrix. The decision is carried out directly on the website (online) and is free. The calculation results are presented in a report in Word and Excel format (i.e., it is possible to check the solution). see design example.
Instructions. To obtain a solution, it is necessary to specify the dimension of the matrix. Next, fill out matrix A in the new dialog box.
See also Inverse matrix using the Jordano-Gauss method
Algorithm for finding the inverse matrix
- Finding the transposed matrix A T .
- Definition of algebraic complements. Replace each element of the matrix with its algebraic complement.
- Compiling 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 whether 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 complements.
- Filling out the union (mutual, adjoint) matrix C .
- Compiling an 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.
- They do a check: they multiply the original and the resulting matrices. The result should be an identity matrix.
Example No. 1. Let's write the matrix in the form:
Algebraic additions.
| A 1,1 = (-1) 1+1 |
|
∆ 1,1 = (-1 4-5 (-2)) = 6
| A 1,2 = (-1) 1+2 |
|
∆ 1,2 = -(2 4-(-2 (-2))) = -4
| A 1.3 = (-1) 1+3 |
|
∆ 1,3 = (2 5-(-2 (-1))) = 8
| A 2,1 = (-1) 2+1 |
|
∆ 2,1 = -(2 4-5 3) = 7
| A 2,2 = (-1) 2+2 |
|
∆ 2,2 = (-1 4-(-2 3)) = 2
| A 2,3 = (-1) 2+3 |
|
∆ 2,3 = -(-1 5-(-2 2)) = 1
| A 3.1 = (-1) 3+1 |
|
∆ 3,1 = (2 (-2)-(-1 3)) = -1
| A 3.2 = (-1) 3+2 |
|
∆ 3,2 = -(-1 (-2)-2 3) = 4
| A 3.3 = (-1) 3+3 |
|
∆ 3,3 = (-1 (-1)-2 2) = -3
Then inverse matrix can be written as:
| A -1 = 1/10 |
|
| A -1 = |
|
Another algorithm for finding the inverse matrix
Let us present another scheme for finding the inverse matrix.- Find the determinant of a given square matrix A.
- We find algebraic complements to all elements of the matrix A.
- We write algebraic additions of row elements to columns (transposition).
- We divide each element of the resulting matrix by the determinant of the matrix A.
A special case: The inverse of the identity matrix E is the identity matrix E.
Definition 1: a matrix is called singular if its determinant is zero.
Definition 2: a matrix is called non-singular if its determinant is not equal to zero.
Matrix "A" is called inverse matrix, if the condition A*A-1 = A-1 *A = E (unit matrix) is satisfied.
A square matrix is invertible only if it is non-singular.
Scheme for calculating the inverse matrix:
1) Calculate the determinant of matrix "A" if ∆ A = 0, then the inverse matrix does not exist.
2) Find all algebraic complements of matrix "A".
3) Create a matrix of algebraic additions (Aij)
4) Transpose the matrix of algebraic complements (Aij )T
5) Multiply the transposed matrix by the inverse of the determinant of this matrix.
6) Perform check:
At first glance it may seem complicated, but in fact everything is very simple. All solutions are based on simple arithmetic operations, the main thing when solving is not to get confused with the “-” and “+” signs and not to lose them.
Now let’s solve a practical task together by calculating the inverse matrix.
Task: find the inverse matrix "A" shown in the picture below:
1. The first thing to do is to find the determinant of matrix "A":
Explanation:
We have simplified our determinant using its basic functions. First, we added to the 2nd and 3rd lines the elements of the first line, multiplied by one number.
Secondly, we changed the 2nd and 3rd columns of the determinant, and according to its properties, we changed the sign in front of it.
Thirdly, we took out the common factor (-1) of the second line, thereby changing the sign again, and it became positive. We also simplified line 3 in the same way as at the very beginning of the example.
We have a triangular determinant whose elements below the diagonal are equal to zero, and by property 7 it is equal to the product of the diagonal elements. In the end we got ∆ A = 26, therefore the inverse matrix exists.
A11 = 1*(3+1) = 4
A12 = -1*(9+2) = -11
A13 = 1*1 = 1
A21 = -1*(-6) = 6
A22 = 1*(3-0) = 3
A23 = -1*(1+4) = -5
A31 = 1*2 = 2
A32 = -1*(-1) = -1
A33 = 1+(1+6) = 7
3. The next step is to compile a matrix from the resulting additions:
5. Multiply this matrix by the inverse of the determinant, that is, by 1/26:
6. Now we just need to check:
During the test, we received an identity matrix, therefore, the solution was carried out absolutely correctly.
2 way to calculate the inverse matrix.
1. Elementary matrix transformation
2. Inverse matrix through an elementary converter.
Elementary matrix transformation includes:
1. Multiplying a string by a number that is not equal to zero.
2. Adding to any line another line multiplied by a number.
3. Swap the rows of the matrix.
4. Applying a chain of elementary transformations, we obtain another matrix.
A -1 = ?
1. (A|E) ~ (E|A -1 )
2.A -1 * A = E
Let's look at this using a practical example with real numbers.
Exercise: Find the inverse matrix.
Solution:
Let's check:
A little clarification on the solution:
First, we rearranged rows 1 and 2 of the matrix, then multiplied the first row by (-1).
After that, we multiplied the first row by (-2) and added it with the second row of the matrix. Then we multiplied line 2 by 1/4.
The final stage of transformation was multiplying the second line by 2 and adding it with the first. As a result, we have the identity matrix on the left, therefore, the inverse matrix is the matrix on the right.
After checking, we were convinced that the decision was correct.
As you can see, calculating the inverse matrix is very simple.
At the end of this lecture, I would also like to spend a little time on the properties of such a matrix.
Methods for finding the inverse matrix, . Consider a square matrix
Let us denote Δ =det A.
The square matrix A is called non-degenerate, or not special, if its determinant is nonzero, and degenerate, or special, IfΔ = 0.
A square matrix B is for a square matrix A of the same order if their product is A B = B A = E, where E is the identity matrix of the same order as the matrices A and B.
Theorem . In order for matrix A to have an inverse matrix, it is necessary and sufficient that its determinant be different from zero.
The inverse matrix of matrix A, denoted by A- 1, so B = A - 1 and is calculated by the formula
, (1)
where A i j are algebraic complements of elements a i j of matrix A..
Calculating A -1 using formula (1) for high-order matrices is very labor-intensive, so in practice it is convenient to find A -1 using the method of elementary transformations (ET). Any non-singular matrix A can be reduced to the identity matrix E by applying only the columns (or only the rows) to the identity matrix. If the transformations perfect over the matrix A are applied in the same order to the identity matrix E, the result will be an inverse matrix. It is convenient to perform EP on matrices A and E simultaneously, writing both matrices side by side through a line. Let us note once again that when searching for the canonical form of a matrix, in order to find it, you can use transformations of rows and columns. If you need to find the inverse of a matrix, you should use only rows or only columns during the transformation process.
Example 2.10. For matrix 
find A -1 .
Solution.First we find the determinant of matrix A 
This means that the inverse matrix exists and we can find it using the formula: 
, where A i j (i,j=1,2,3) are algebraic additions of elements a i j of the original matrix. 
Where 
.
Example 2.11. Using the method of elementary transformations, find A -1 for the matrix: A = .
Solution.We assign to the original matrix on the right an identity matrix of the same order: 
. Using elementary transformations of the columns, we will reduce the left “half” to the identity one, simultaneously performing exactly the same transformations on the right matrix.
To do this, swap the first and second columns: 
~
. To the third column we add the first, and to the second - the first, multiplied by -2: 
. From the first column we subtract the second doubled, 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 matrix obtained to the right of the vertical bar is the inverse matrix of the given matrix A. So,
.
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