Matrix transposition in Microsoft Excel. Matrix transposition online Matrix transport online

The matrix А -1 is called the inverse matrix with respect to the matrix А if А * А -1 = Е, where Е is the n-th order unit matrix. An inverse matrix can only exist for square matrices.

Service purpose... With the help of this service online, you can find algebraic complements, transposed matrix A T, adjoint matrix and inverse matrix. The solution is carried out directly on the website (online) and is free of charge. The calculation results are presented in a Word report and in Excel format (i.e. it is possible to check the solution). see design example.

Instruction. To obtain a solution, it is necessary to set the dimension of the matrix. Next, in a new dialog box, fill in the matrix A.

See also Inverse matrix using the Jordan-Gauss method

Algorithm for finding the inverse matrix

1. Finding the transposed matrix A T.
2. Definition of algebraic complements. Replace each element of the matrix with its algebraic complement.
3. 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.

Next inverse matrix algorithm is similar to the previous one, except for some steps: first, the algebraic complements are calculated, and then the adjoint matrix C is determined.

1. Determine if the matrix is ​​square. If not, then there is no inverse matrix for it.
2. 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.
3. Definition of algebraic complements.
4. Filling the union (reciprocal, adjoint) matrix C.
5. Composing an inverse matrix from algebraic complements: 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.
6. A check is made: the original and the resulting matrices are multiplied. The result should be the identity matrix.

Example # 1. Let's write the matrix as follows:

Algebraic complements. ∆ 1,2 = - (2 4 - (- 2 (-2))) = -4 ∆ 2,1 = - (2 4-5 3) = 7 ∆ 2,3 = - (- 1 5 - (- 2 2)) = 1 ∆ 3.2 = - (- 1 (-2) -2 3) = 4

A -1 =

0,6 -0,4 0,8 0,7 0,2 0,1 -0,1 0,4 -0,3

Another algorithm for finding the inverse matrix

Let us give another scheme for finding the inverse matrix.

1. Find the determinant of the given square matrix A.
2. Find the algebraic complements to all elements of the matrix A.
3. We write the algebraic complements of row elements into columns (transposition).
4. We divide each element of the resulting matrix by the determinant of the matrix A.

As you can see, the transposition operation can be applied both at the beginning, over the original matrix, and at the end, over the obtained algebraic complements.

A special case: The inverse of the identity matrix E is the identity matrix E.

Transposing a matrix through this online calculator will not take you much time, but it will quickly give a result and help you better understand the process itself.

Sometimes in algebraic calculations there is a need to swap the rows and columns of a matrix. This operation is called matrix transposition. The rows become columns in order, and the matrix itself becomes transposed. There are certain rules in these calculations, and in order to understand them and visually familiarize yourself with the process, use this online calculator. It will greatly facilitate your task and help you better understand the theory of matrix transposition. A significant advantage of this calculator is the demonstration of a detailed and detailed solution. Thus, its use contributes to obtaining a deeper and more informed understanding of algebraic calculations. In addition, with its help you can always check how successfully you have coped with the task by manually transposing the matrices.

The calculator is very easy to use. To find the transposed matrix online, indicate the size of the matrix by clicking on the "+" or "-" icons until the desired values ​​for the number of columns and rows are obtained. Next, the required numbers are entered into the fields. Below is the button "Calculate" - pressing it displays a ready-made solution with a detailed explanation of the algorithm.

To transpose a matrix, you need to write the rows of the matrix into columns.

If, then the transposed matrix

If, then

Exercise 1. Find

1. Determinants of square matrices.

For square matrices, a number is entered, which is called a determinant.

For matrices of the second order (dimension), the determinant is given by the formula:

For example, for a matrix, its determinant

Example . Calculate determinants of matrices.

For square matrices of the third order (dimension), there is a "triangle" rule: in the figure, the dotted line means - multiply the numbers through which the dotted line passes. The first three numbers must be added, the next three numbers must be subtracted.

Example... Calculate the determinant.

To give a general definition of a determinant, it is necessary to introduce the concept of a minor and an algebraic complement.

Minor the element of the matrix is ​​called the determinant obtained by deleting - that row and - that column.

Example. Let us find some minors of the matrix A.

Algebraic complement element is called a number.

This means that if the sum of the indices and is even, then they are no different. If the sum of the indices and is odd, then they differ only in sign.

For the previous example.

The determinant of the matrix is the sum of the products of elements of some string

(column) by their algebraic complements. Consider this definition on a third-order matrix.

The first record is called the factorization of the determinant in the first row, the second is the factorization in the second column, and the last is the decomposition in the third row. In total, such expansions can be written six times.

Example... Calculate the determinant according to the "triangle" rule and expanding it along the first line, then along the third column, then along the second line.

Let's expand the determinant along the first line:

Let us expand the determinant by the third column:

Let's expand the determinant along the second line:

Note that the more zeros there are, the easier the calculations are. For example, expanding along the first column, we get

Among the properties of determinants there is a property that allows getting zeros, namely:

If to the elements of a certain row (column) we add the elements of another row (column) multiplied by a nonzero number, then the determinant will not change.

Let's take the same determinant and get zeros, for example, in the first line.

Higher order determinants are calculated in the same way.

Task 2. Calculate the fourth order determinant:

1) expanding to any row or any column

2) having previously received zeros

We get an additional zero, for example, in the second column. To do this, multiply the elements of the second line by -1 and add to the fourth line:

1. Solving systems of linear algebraic equations by Cramer's method.

Let us show the solution of a system of linear algebraic equations by Cramer's method.

Task 2. Solve the system of equations.

It is necessary to calculate four determinants. The first is called the main one and consists of the coefficients for the unknowns:

Note that if, the system cannot be solved by Cramer's method.

The other three determinants are denoted,, and are obtained by replacing the corresponding column with the column of the right-hand sides.

We find. To do this, we change the first column in the main determinant to the column of the right-hand sides:

We find. To do this, change the second column in the main determinant to the column of the right-hand sides:

We find. To do this, change the third column in the main determinant to the column of the right-hand sides:

We find the solution of the system by Cramer's formulas:,,

Thus, the solution of the system,

Let's make a check, for this we substitute the found solution into all the equations of the system.

1. Solution of systems of linear algebraic equations by the matrix method.

If a square matrix has a nonzero determinant, there is an inverse matrix such that. The matrix is ​​called unit and has the form

The inverse matrix is ​​found by the formula:

Example... Find the inverse of a matrix

First, we compute the determinant.

Find algebraic complements:

We write the inverse matrix:

To check the calculations, you need to make sure that.

Let a system of linear equations be given:

We denote

Then the system of equations can be written in matrix form as, and hence. The resulting formula is called the matrix method for solving the system.

Task 3. Solve the system in a matrix way.

It is necessary to write out the matrix of the system, find its inverse and then multiply by the column of the right-hand sides.

We have already found the inverse matrix in the previous example, so we can find a solution:

1. Solution of systems of linear algebraic equations by the Gauss method.

Cramer's method and matrix method are used only for quadratic systems (the number of equations is equal to the number of unknowns), and the determinant must not be equal to zero. If the number of equations is not equal to the number of unknowns, or the determinant of the system is zero, the Gaussian method is applied. Gauss's method can be used to solve any system.

And substitute in the first equation:

Task 5. Solve the system of equations by the Gauss method.

Using the resulting matrix, we restore the system:

We find a solution:

When working with matrices, sometimes you need to transpose them, that is, in simple words, turn them over. Of course, you can kill the data manually, but Excel offers several ways to make it easier and faster. Let's take a look at them in detail.

Matrix transpose is the process of swapping columns and rows. Excel has two options for transposing: using the function TRANSPOSE and using the special insert tool. Let's consider each of these options in more detail.

Method 1: operator TRANSPOSE

Function TRANSPOSE belongs to the category of operators References and Arrays... A peculiarity is that, like other functions that work with arrays, the result of issuing is not the contents of the cell, but a whole array of data. The syntax for the function is pretty simple and looks like this:

TRANSPOSE (array)

That is, the only argument of this operator is a reference to the array, in our case the matrix, which should be transformed.

Let's see how this function can be applied using an example with a real matrix.

1. Select the empty cell on the sheet, which is planned to be the upper-leftmost cell of the transformed matrix. Next, click on the icon "Insert function" which is located near the formula bar.



2. Launch in progress Function Wizards... We open a category in it References and Arrays or "Complete alphabetical listing"... After finding the name "TRANSP", select it and click on the button "OK".



3. The function arguments window is launched. TRANSPOSE... The only argument of this operator corresponds to the field "Array"... You need to enter the coordinates of the matrix into it, which should be turned over. To do this, place the cursor in the field and, holding down the left mouse button, select the entire range of the matrix on the sheet. After the address of the area is displayed in the arguments window, click on the button "OK".



4. But, as you can see, in the cell, which is intended for displaying the result, an incorrect value is displayed in the form of an error "#VALUE!"... This is due to the way the array operators work. To fix this error, select a range of cells in which the number of rows should be equal to the number of columns in the original matrix, and the number of columns should be equal to the number of rows. This correspondence is very important in order for the result to be displayed correctly. Moreover, the cell containing the expression "#VALUE!" should be the upper left cell of the selected array and it is from this cell that the selection procedure should be started by holding down the left mouse button. After you have made a selection, place the cursor on the formula bar immediately after the statement of the operator TRANSPOSE, which should be displayed in it. After that, to make the calculation, you need to click not on the button Enter, as is customary in usual formulas, and dial the combination Ctrl + Shift + Enter.



5. After these actions, the matrix was displayed as we need it, that is, in a transposed form. But there is another problem. The fact is that now the new matrix is ​​an array bound by the formula, which cannot be changed. When you try to make any change with the contents of the matrix, an error will pop up. Some users are quite satisfied with this state of affairs, since they are not going to make changes in the array, but others need a matrix with which they can fully work.

   To solve this problem, select the entire transposed range. By moving to the tab "Home" click on the icon "Copy", which is located on the ribbon in the group "Clipboard"... Instead of the specified action, after selection, you can make a set of standard keyboard shortcuts for copying Ctrl + C.



6. Then, without removing the selection from the transposed range, click on it with the right mouse button. In the context menu in the group Paste Options click on the icon "Values", which looks like a pictogram depicting numbers.

   

   This is followed by the array formula TRANSPOSE will be deleted, and only one values ​​will remain in the cells, with which you can work in the same way as with the original matrix.

Method 2: transpose a matrix using paste special

In addition, the matrix can be transposed using one item of the context menu, which is called "Paste special".

After these actions, only the transformed matrix will remain on the sheet.

In the same two ways, which were discussed above, you can transpose in Excel not only matrices, but also full-fledged tables. The procedure will be almost identical.

So, we found out that in Excel, a matrix can be transposed, that is, flipped by changing columns and rows in two ways. The first option involves using the function TRANSPOSE and the second is the Paste Special Tools. By and large, the end result that is obtained using both of these methods is no different. Both methods work in almost any situation. So when choosing a conversion option, the personal preferences of a particular user come to the fore. That is, which of these methods is more convenient for you personally, use that one.