Methods for constructing an inverse matrix. Method of elementary transformations (Gauss and Gauss-Jordan methods for finding inverse matrices)

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.

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 built matrix method solving systems of equations. Our service site allows calculate inverse matrix online two methods: the Gauss-Jordan method and using the matrix of algebraic additions. Interrupt implies a large number of elementary transformations inside the matrix, the second is 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

Find the inverse matrix for the site

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In order to find the inverse matrix online, you will need to indicate the size of the matrix itself. To do this, click on the “+” or “-” icons until you are satisfied with the number of columns and rows. Next, enter the required elements in the fields. Below is the “Calculate” button - by clicking it, you will receive an answer on the screen with a detailed solution.

In linear algebra, quite often one has to deal with the process of calculating the inverse matrix. It exists only for unexpressed matrices and for square matrices provided that the determinant is nonzero. In principle, calculating it is not particularly difficult, especially if you are dealing with a small matrix. But if you need more complex calculations or carefully double-check your decision, it is better to use this online calculator. With its help, you can quickly and accurately solve an inverse matrix.

Using this online calculator you can make your calculations much easier. In addition, it helps to consolidate the material obtained in theory - it is a kind of simulator for the brain. It should not be considered as a replacement for manual calculations; it can give you much more, making it easier to understand the algorithm itself. Besides, it never hurts to double-check yourself.

inverse matrix

Inverse matrix is a matrix that, when multiplied both on the right and on the left by a given matrix, gives the identity matrix.
Let us denote the inverse matrix of the matrix A through , then according to definition we get:

Where E– identity matrix.
Square matrix called not special (non-degenerate) if its determinant is not zero. Otherwise it is called special (degenerate) or singular.

The theorem holds: Every non-singular matrix has an inverse matrix.

The operation of finding the inverse matrix is ​​called appeal matrices. Let's consider the matrix inversion algorithm. Let a non-singular matrix be given n-th order:

where Δ = det A ≠ 0.

Algebraic addition of an element matrices n-th order A is called the determinant of a matrix taken with a certain sign ( n–1)th order obtained by deleting i-th line and j th matrix column A:

Let's create the so-called attached matrix:

where are the algebraic complements of the corresponding elements of the matrix A.
Note that algebraic additions of matrix row elements A are placed in the corresponding columns of the matrix à , that is, the matrix is ​​transposed at the same time.
By dividing all the elements of the matrix à by Δ – the value of the matrix determinant A, we get the inverse matrix as a result:

Let us note a number of special properties of the inverse matrix:
1) for a given matrix A its inverse matrix is the only one;
2) if there is an inverse matrix, then right reverse And left reverse the matrices coincide with it;
3) a special (singular) square matrix does not have an inverse matrix.

Basic properties of an inverse matrix:
1) the determinant of the inverse matrix and the determinant of the original matrix are reciprocals;
2) the inverse matrix of the product of square matrices is equal to the product inverse matrices factors taken in reverse order:

3) the transposed inverse matrix is ​​equal to the inverse matrix of the given transposed matrix:

EXAMPLE Calculate the inverse of the given matrix.

Typically, inverse operations are used to simplify complex algebraic expressions. For example, if the problem involves the operation of dividing by a fraction, you can replace it with the operation of multiplying by the reciprocal of a fraction, 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 using a good graphing calculator.

Steps

Using the adjoint 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 change rows to 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.