As a matrix is \u200b\u200berected into a square. Finding the reverse matrix. Operations on matrices and their properties
Here we will continue to be launched in the first part of the operations over the matrices and wonder the pair of examples in which you will need to apply several operations at once.
Construction of the matrix to the degree.
Let K be a non-negative number. For any square matrix $ A_ (N \\ Times N) $ we have: $$ a ^ k \u003d \\ underbrace (a \\ cdot a \\ cdot \\ ldots \\ cdot a) _ (k \\; times) $$
In this case, we assume that $ A ^ 0 \u003d E $, where $ e $ is a single matrix of the corresponding order.
Example number 4.
The matrix $ a \u003d \\ left (\\ begin (array) (CC) 1 & 2 \\\\ -1 & -3 \\ END (Array) \\ Right) $ is set. Find the matrices $ a ^ 2 $ and $ a ^ $ 6.
According to the definition of $ a ^ 2 \u003d A \\ Cdot A $, i.e. To find $ a ^ $ 2 $ we just need to multiply the $ a $ matrix for yourself. The multiplication operation of the matrices was considered in the first part of the topic, so here we simply write the solution process without detailed explanations:
$$ A ^ 2 \u003d A \\ CDOT A \u003d \\ LEFT (\\ Begin (Array) (CC) 1 & 2 \\\\ -1 & -3 \\ END (Array) \\ Right) \\ CDOT \\ Left (\\ Begin (Array) (CC) 1 & 2 \\\\ -1 & -3 \\ END (Array) \\ Right) \u003d \\ Left (\\ Begin (Array) (CC) 1 \\ CDOT 1 + 2 \\ CDOT (-1) & 1 \\ CDOT 2 +2 \\ Cdot (-3) \\\\ -1 \\ CDOT 1 + (- 3) \\ CDot (-1) & -1 \\ CDOT 2 + (- 3) \\ CDOT (-3) \\ END (Array) \\ Right ) \u003d \\ left (\\ begin (array) (CC) -1 & -4 \\\\ 2 & 7 \\ END (Array) \\ Right). $$.
To find the $ a ^ $ 6 matrix we have two options. Option FIRST: tritely continue to multiply $ a ^ $ 2 on a $ a $ matrix:
$$ a ^ 6 \u003d a ^ 2 \\ Cdot A \\ Cdot A \\ Cdot A \\ Cdot A. $$
However, it is possible to go somewhat simpler through, using the properties of the associativity of multiplication of matrices. We put brackets in the expression for $ a ^ $ 6:
$$ a ^ 6 \u003d a ^ 2 \\ Cdot a \\ cdot a \\ cdot a \\ cdot a \u003d a ^ 2 \\ Cdot (a \\ cdot a) \\ CDot (A \\ Cdot a) \u003d a ^ 2 \\ Cdot A ^ 2 \\ Cdot a ^ 2. $$.
If, when solving the first method, there would be four multiplication operations, then for the second method - only two. So let's go through the second way:
$$ a ^ 6 \u003d a ^ 2 \\ Cdot A ^ 2 \\ Cdot A ^ 2 \u003d \\ Left (\\ Begin (Array) (CC) -1 & -4 \\\\ 2 & 7 \\ END (Array) \\ Right) \\ 2 & 7 \\ END (Array) \\ Right) \u003d \\\\ \u003d \\ Left (\\ Begin (Array) (CC) -1 \\ CDOT (-1) + (- 4) \\ CDOT 2 & -1 \\ CDOT (-4 ) + (- 4) \\ Cdot 7 \\\\ 2 \\ Cdot (-1) +7 \\ CDOT 2 & 2 \\ CDOT (-4) +7 \\ CDOT 7 \\ END (Array) \\ Right) \\ Cdot \\ Left (\\ Array) \\ Right) \\ Cdot \\ Left (\\ Begin (Array) (CC) -1 & -4 \\\\ 2 & 7 \\ END (Array) \\ Right) \u003d \\\\ \u003d \\ Left (\\ Begin (Array) (CC ) -7 \\ Cdot (-1) + (- 24) \\ CDOT 2 & -7 \\ Cdot (-4) + (- 24) \\ CDOT 7 \\\\ 12 \\ CDOT (-1) +41 \\ CDOT 2 & 12 \\ CDOT (-4) +41 \\ CDOT 7 \\ END (Array) \\ Right) \u003d \\ Left (\\ Begin (Array) (CC) -41 & -140 \\\\0 & 239 \\ END (Array) \\ Right). $$.
Answer: $ A ^ 2 \u003d \\ Left (\\ Begin (Array) (CC) -1 & -4 \\\\ 2 & 7 \\ END (Array) \\ Right) $, $ a ^ 6 \u003d \\ left (\\ Begin (Array) (CC) -41 & -140 \\\\0 & 239 \\ END (Array) \\ Right) $.
Example number 5.
Matrix $ A \u003d \\ Left (\\ Begin (Array) (CCCC) 1 & 0 & -1 & 2 \\\\ 3 & 4 & 5 & 0 \\ ED (Array) \\ Right) $, $ B \u003d \\ Left (\\ Begin (Array) (CCC) -9 & 1 & 0 \\\\ 2 & -1 & 4 \\\\ 0 & -2 & 3 \\\\ 1 & 5 & 0 \\ END (Array) \\ Right) $, $ C \u003d \\ LEFT (\\ Begin (Array) (CCC) -5 & -20 & 13 \\\\ 10 & 12 & 9 \\\\ 3 & -15 & 8 \\ END (Array) \\ Find the matrix $ d \u003d 2ab-3c ^ T + 7e $.
Calculation of the matrix of the $ D $ will start with finding the result of the product $ AB $. The matrices $ a $ and $ b $ can be multiplied, since the number of columns of the $ a $ matrix column is equal to the number of lines of the matrix $ b $. Denote by $ F \u003d AB $. In this case, the matrix $ F will have three columns and three lines, i.e. It will be square (if this output seems unclear, see the description of multiplication of matrices in the first part of this topic). We find the $ F $ matrix, calculates all its elements:
$$ F \u003d A \\ CDOT B \u003d \\ LEFT (\\ Begin (Array) (CCCC) 1 & 0 & -1 & 2 \\\\ 3 & -2 & 5 & 0 \\\\ -1 & 4 & -3 & 6 \\ End (Array) \\ Right) \\\\ \\ Begin (Aligned) & F_ (11) \u003d 1 \\ CDOT (-9) +0 \\ CDOT 2 + (- 1) \\ CDOT 0 + 2 \\ CDOT 1 \u003d -7; \\\\ & F_ (12) \u003d 1 \\ CDot 1 + 0 \\ Cdot (-1) + (- 1) \\ Cdot (-2) +2 \\ Cdot 5 \u003d 13; \\\\ & F_ (13) \u003d 1 \\ Cdot 0 + 0 \\ Cdot 4 + (- 1) \\ Cdot 3 + 2 \\ Cdot 0 \u003d -3; \\\\ \\\\ & F_ (21) \u003d 3 \\ CDOT (-9 ) + (- 2) \\ CDot 2 + 5 \\ Cdot 0 + 0 \\ Cdot 1 \u003d -31; \\\\ & F_ (22) \u003d 3 \\ CDOT 1 + (- 2) \\ CDot (-1) +5 \\ Cdot (-2) +0 \\ CDot 5 \u003d -5; \\\\ & F_ (23) \u003d 3 \\ CDot 0 + (- 2) \\ CDot 4 + 5 \\ Cdot 3 + 0 \\ Cdot 0 \u003d 7; \\\\ \\\\ \\\\ & F_ (32) \u003d - 1 \\ CDot 1 + 4 \\ Cdot (-1) + (- 3) \\ Cdot (-2) +6 \\ Cdot 5 \u003d 31; \\\\ & F_ (33) \u003d - 1 \\ Cdot 0 + 4 \\ Cdot 4 + (- 3) \\ Cdot 3 + 6 \\ Cdot 0 \u003d 7. \\ End (Aligned) $$
So, $ F \u003d \\ Left (\\ Begin (Array) (CCC) -7 & 13 & -3 \\\\ -31 & -5 & 7 \\\\ 23 & 31 & 7 \\ END (Array) \\ Right) $. Let's go further. Matrix $ C ^ T $ - transposed matrix for a $ C $ matrix, i.e. $ C ^ T \u003d \\ Left (\\ Begin (Array) (CCC) -5 & 10 & 3 \\\\ -20 & 12 & -15 \\\\ 13 & 9 & 8 \\ END (Array) \\ Right) $. As for the matrix $ E $, then this is a single matrix. In this case, the order of this matrix is \u200b\u200bthree, i.e. $ E \u003d \\ LEFT (\\ Begin (Array) (CCC) 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\ End (Array) \\ Right) $.
In principle, we can continue to go step by step, but the remaining expression is better to consider entirely without being distracted by auxiliary actions. In fact, we have only operations for multiplication of matrices for a number, as well as operations of addition and subtraction.
$$ d \u003d 2ab-3c ^ t + 7e \u003d 2 \\ CDOT \\ LEFT (\\ Begin (Array) (CCC) -7 & 13 & -3 \\\\ -31 & -5 & 7 \\\\ 23 & 31 & 7 \\ Right) +7 \\ Cdot \\ Left (\\ Begin (Array) (CCC) 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\ End (Array) \\ Right) $$
Multiply matrices in the right part of equality on the corresponding numbers (i.e., 2, 3 and 7):
$$ 2 \\ CDOT \\ LEFT (\\ Begin (Array) (CCC) -7 & 13 & -3 \\\\ -31 & -5 & 7 \\\\ 23 & 31 & 7 \\ END (Array) \\ Right) -3 \\ Begin (Array) (CCC) 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\ END (Array) \\ Right) \u003d \\\\ \u003d \\ LEFT (\\ Begin (Array) (CCC) - 14 & 26 & 14 \\\\ -62 & -10 & 14 \\\\ 46 & 62 & 14 \\ END (Array) \\ Right) - \\ Left (\\ Begin (Array) (CCC) -15 & 13 & 9 \\\\ 0 & 7 \\ END (Array) \\ Right) $$
Perform the latest actions: subtraction and addition:
$$ \\ Left (\\ Begin (Array) (CCC) -14 & 26 & -6 \\\\ -62 & -10 & 14 \\\\6 & 62 & 14 \\ END (Array) \\ Right) - \\ Left (\\ Begin (Array) (CCC) -15 & 30 & 9 \\\\ -60 & 36 & -45 \\\\ 39 & 27 & 24 \\ END (Array) \\ Right) + \\ Left (\\ Begin (Array) (CCC) 7 & 0 & 0 \\\\ 0 & 7 \\ END (Array) \\ Right) \u003d \\\\ \u003d \\ Left (\\ Begin (Array) (CCC) -14 - (- 15) +7 & 26-30 + 0 & -6-9 + 0 \\\\ -62 - (- 60) +0 & -10-36 + 7 & 14 - (- 45) +0 \\\\ 46-39 + 0 & 62-27 +0 & 14-24 + 7 \\ END (Array) \\ Right) \u003d \\ Left (\\ Begin (Array) (CCC) 8 & -4 & -15 \\\\ -2 & -39 & 59 \\\\ 7 & 35 & -3 \\ END (Array) \\ Right). $$.
The task is solved, $ d \u003d \\ left (\\ begin (array) (CCC) 8 & -4 & -15 \\\\ -2 & -39 & 59 \\\\ 7 & 35 & -3 \\ End (Array) \\ Right) $ .
Answer: $ D \u003d \\ left (\\ begin (array) (CCC) 8 & -4 & -15 \\\\ -2 & -39 & 59 \\\\ 7 & 35 & -3 \\ END (Array) \\ Right) $.
Example number 6.
Let $ f (x) \u003d 2x ^ 2 + 3x-9 $ and the matrix $ a \u003d \\ left (\\ begin (array) (CC) -3 & 1 \\\\ 5 & 0 \\ END (Array) \\ Right) $. Find the value of $ F (a) $.
If $ f (x) \u003d 2x ^ 2 + 3x-9 $, then under $ F (a) $ understand the matrix:
$$ F (a) \u003d 2a ^ 2 + 3a-9e. $$.
This is how the polynomial is determined from the matrix. So, we need to substitute the matrix $ a $ in the expression for $ F (a) $ and get the result. Since all actions were dismantled in detail earlier, then I'll just give a decision. If the process of execution of the operation $ a ^ 2 \u003d a \\ cdot a $ is unclear for you, I advise you to look at the description of multiplication of matrices in the first part of this topic.
$$ F (a) \u003d 2a ^ 2 + 3a-9e \u003d 2a \\ Cdot A + 3A-9E \u003d 2 \\ Left (\\ Begin (Array) (CC) -3 & 1 \\\\ 5 & 0 \\ END (Array) \\ Right) \\ CDOT \\ Left (\\ Begin (Array) (CC) -3 & 1 \\\\ 5 & 0 \\ END (Array) \\ Right) +3 \\ Left (\\ Begin (Array) (CC) -3 & 1 \\\\ 5 & 0 \\ END (Array) \\ Right) -9 \\ Left (\\ Begin (Array) (CC) 1 & 0 \\\\ 0 & 1 \\ END (Array) \\ Right) \u003d \\\\ \u003d 2 \\ left ( \\ Begin (Array) (CC) (-3) \\ CDOT (-3) +1 \\ CDOT 5 & (-3) \\ CDOT 1 + 1 \\ CDOT 0 \\\\ 5 \\ CDOT (-3) +0 \\ CDOT 5 & 5 \\ CDot 1 + 0 \\ Cdot 0 \\ END (Array) \\ Right) +3 \\ Left (\\ Begin (Array) (CC) -3 & 1 \\\\ 5 & 0 \\ END (Array) \\ Right) -9 \\ Left (\\ Begin (Array) (CC) 1 & 0 \\\\ 0 & 1 \\ END (Array) \\ Right) \u003d \\\\ \u003d 2 \\ Left (\\ Begin (Array) (CC) 14 & -3 \\\\ - 15 & 5 \\ END (Array) \\ Right) +3 \\ Left (\\ Begin (Array) (CC) -3 & 1 \\\\ 5 & 0 \\ END (Array) \\ Right) -9 \\ Left (\\ Begin (Array ) (CC) 1 & 0 \\\\ 0 & 1 \\ END (Array) \\ Right) \u003d \\ Left (\\ Begin (Array) (CC) 28 & -6 \\\\ -30 & 10 \\ END (Array) \\ Right) + \\ LEFT (\\ Begin (Array) (CC) -9 & 3 \\\\ 15 & 0 \\ END (Array) \\ Right) - \\ Left (\\ Begin (Array) (CC) 9 & 0 \\\\ 0 & 9 \\ $$.
Answer: $ F (a) \u003d \\ Left (\\ Begin (Array) (CC) 10 & -3 \\\\ -15 & 1 \\ END (Array) \\ Right) $.
It should be noted that only square matrices can be given. Equal number of rows and columns - a prerequisite for the construction of the matrix to the degree. During the calculation, the matrix will be multiplied by the required number of times.
This online calculator is designed to perform a matrix erection operation. Thanks to its use, you will not only quickly cope with this task, but also get a visual and deployment of the progress of the progress. This will help better consolidate the material obtained in the theory. Seeing the detailed algorithm of calculations, you will better understand all its subtleties and can subsequently do not allow errors in manual calculation. In addition, it will never be superfluous to double-check their calculations, and it is also best to exercise here.
In order to build a matrix into an online degree, you will need a number of simple actions. First of all, specify the size of the matrix by clicking on the "+" or "-" icons to the left of it. Then enter the numbers in the matrix field. You also need to specify the degree in which the matrix is \u200b\u200berected. And then you can only click on the button: "Calculate" at the bottom of the field. The result obtained will be reliable and accurate if you carefully and correctly entered all values. Together with him you will be provided with a detailed decoding solution.
Linear algebra for teapots
To study a linear algebra, you can read and delve into the book I. V. Belousov "Matrixes and Deterpetes". However, it is written with a strict and dry mathematical language, which people with the Middle mind take hard. Therefore, I made a retelling of the most difficult for understanding the places of this book, trying to state the material as clearer as possible, using the drawings as much as possible. The proofs of the theorems I lowered. To admit, I myself did not understand them. Believe Mr. Belousov! Judging by his work, he is a competent and sensible mathematician. You can download his book at http://eqworld.ipmnet.ru/ru/library/books/belousov2006ru.pdf.If you're going to delve into my work, it needs to be done because I will often refer to Belousov.
Let's start with definitions. What is a matrix? This is a rectangular table of numbers, functions or algebraic expressions. Why do you need matrices? They greatly facilitate complex mathematical calculations. The matrix uses strings and columns (Fig. 1).
Rows and columns are numbered, starting on the left
from above (Fig. 1-1). When they say: the matrix of the size M n (or m per n) is implied under m Number of string, and under n Number of columns. For example, the matrix in Figure 1-1 has the size "4 to 3", and not "3 to 4".
See in fig. 1-3, what are the matrices. If the matrix consists of one line, it is called a string matrix, and if from one column, then a column matrix. The matrix is \u200b\u200bcalled a square N-th order if the number of rows is equal to the number of columns and equal to N. If all the matrix elements are zero, then this is a zero matrix. The square matrix is \u200b\u200bcalled diagonal if zero is equal to all its elements, except those located on the main diagonal.
Immediately explain what is the main diagonal. On it numbers rows and columns are the same. It goes from left to right from top to bottom. (Fig. 3) Elements are called diagonal if they are located on the main diagonal. If all the diagonal elements are equal to one (and the remaining zero), the matrix is \u200b\u200bcalled a single one. Two matrices A and B are the same size called equal, if all their elements are the same.
2 Operations on matrices and their properties
The work of the matrix to the number X is the matrix of the same size. To get this product, you need to multiply each element to this number (Fig. 4). To obtain the sum of two matrices of the same size, you need to add their corresponding elements (Fig. 4). To obtain the difference A - B of two matrices of the same size, you need to multiply the matrix B to -1 and add the resulting matrix with the matrix A (Fig. 4). For operations on matrices, properties are valid: a + B \u003d B + A (commutative property).
(A + B) + C \u003d A + (B + C) (Associativity property). By simple, speaking, the amount does not change from the change of places. For operations on matrices and numbers, properties are valid:
(Denote by the number of letters x and y, and the matrix letters a and b) x (ya) \u003d (xy) a
These properties are similar to the properties acting on operations over numbers. See
examples Figure 5. Also, see examples 2.4 - 2.6 Belousov on page 9.
Matrix multiplication.
The multiplication of two matrices is defined only then (translated into Russian: matrices can be multiplied only then) when the number of columns of the first matrix in the work is equal to the number of strings of the second (Fig. 7, at the top, blue brackets). To better remember: the figure 1 is more like a column.As a result of multiplication, a matrix of size is obtained (see Figure 6). To make it easier to remember what you need to multiply, I propose the following algorithm: we look at Figure 7. We multiply the matrix A on the matrix B.
matrix A two columns,
in the matrix b two lines - you can multiply.
1) We will deal with the first column of the matrix B (it only has it only). We write this column in the string (we transpose
column, about transposing just below).
2) Copy this string so that we have a matrix with a matrix of A.
3) Multiply the elements of this matrix to the corresponding elements of the matrix A.
4) Fold the resulting works in each line and getmatrix-work of two lines and one column.
Figure 7-1 shows examples of multiplication of matrices that are more than whiter.
1) Here at the first matrix three columns, it means that the second must have three lines. The algorithm is exactly the same that in the previous example, only here in each line three terms, and not two.
2) Here the second matrix has two columns. First, we do the algorithm with the first column, then with the second, and we get the "two two" matrix.
3) Here at the second matrix, the column consists of one element, the column will not change from transposition. And it is not necessary to put anything, because in the first matrix only one column. We do the algorithm three times and get the "three three" matrix.
The following properties take place:
1. If the sum B + C and the AB product exist, then A (B + C) \u003d AB + AC
2. If the AB product exists, x (ab) \u003d (xa) b \u003d a (xb).
3. If the works of AB and BC exist, then A (BC) \u003d (AB) C.
If the product of AB matrices exists, then the product Ba may not exist. Even the works of AB and BA exist, they may be matrices of different sizes.
Both works of AB and Ba exist and are matrices of the same size only in the case of square matrices A and B of the same order. However, even in this case, AB may not be equal to BA.
Erend into degree
The construction of the matrix into a degree makes sense only for square matrices (think about why?). Then the whole positive degree M matrix A is the product of M matrices equal to A. Just like the numbers. Under the zero degree of a square matrix A is a single matrix of the same order as A. if forgot what a single matrix is, look in Fig. 3.
Also, as in numbers, the following ratios take place:
A ma k \u003d a m + k (a m) k \u003d a mk
See the examples of Belousov on page 20.
Transposing matrices
Transposition - This conversion of the matrix A in the AT matrix,
in which the strings of the matrix A are recorded in the AT columns while preserving the order. (Fig. 8). You can say differently:
the columns of the matrix A are recorded in the AT matrix rows with the preservation of order. Note that when transposing the matrix size changes, that is, the number of rows and columns. Also note that elements on the first line, first column, and the last line, last column remain in place.
The following properties take place: (AT) T \u003d A (transponder
the matrix twice - you will get the same matrix)
(xa) t \u003d xat (under x meant the number, under a, of course, the matrix) (if you need to multiply the matrix to the number and transpose, you can first multiply, then transpose, and you can vice versa)
(A + B) T \u003d AT + BT (AB) T \u003d BT AT
Symmetric and antisymmetric matrices
Figure 9 at the top of the left shows a symmetric matrix. Its elements, symmetrical relative to the main diagonal, are equal. And now Definition: Square Matrix
A is called symmetric if at \u003d a. That is, the symmetric matrix during transpose does not change. In particular, symmetric is any diagonal matrix. (Such a matrix is \u200b\u200bdepicted in Fig. 2).
Now look at the antisymmetric matrix (Fig. 9, bottom). What does it differ from symmetrical? Please note that all its diagonal elements are zero. In antisymmetric matrices, all diagonal elements are zero. Think why? Definition: Square Matrix A is called
antisymmetric, if at \u003d -a. Note some properties of operations over symmetrical and antisymmetric
matrians. 1. If A and B are symmetrical (antisymmetric) matrices, then A + B is a symmetric (antisymmetric) matrix.
2.If a - symmetric (antisymmetric) matrix, then Xa is also a symmetric (antisymmetric) matrix. (In fact, if you multiply the matrix from the figure 9 to some number, the symmetry will still be saved)
3. The product of AB of two symmetric or two antisymmetric matrices A and B is a matrix symmetrical with AB \u003d BA and antisymmetric with AB \u003d-Ba.
4. If a is a symmetric matrix, thenm (m \u003d 1, 2, 3,...) - Symmetric matrix. If A.
Antisymmetric matrix, then AM (m \u003d 1, 2, 3,...) It is a symmetric matrix with even M and antisymmetric - with odd.
5. An arbitrary square matrix A can be represented as the sum of two matrices. (Let's call these matrices, for example A (S) and A (A))
A \u003d a (s) + a (a)
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One of these code options must be copied and insert into the code of your webpage, preferably between the tags
and or immediately after the tag . According to the first version, MathJax is loaded faster and slows down the page. But the second option automatically tracks and loads the latest MathJax versions. If you insert the first code, it will need to be periodically updated. If you insert the second code, the pages will be loaded slower, but you will not need to constantly monitor MathJax updates.Connect MathJax is the easiest way to Blogger or WordPress: Add a widget for inserting a third-party JavaScript code to insert the first or second version of the download code presented above and place the widget closer to the beginning of the template (by the way, it is not at all necessary Since the MathJax script is loaded asynchronously). That's all. Now read Mathml, Latex and Asciimathml markup syntax, and you are ready to insert mathematical formulas on the web pages of your site.
Another New Year's Eve ... Frosty Weather and Snowflakes on the window glass ... All this prompted me to again write about ... Fractals, and about what he knows about this Alpha Tougar. On this occasion, there is an interesting article in which there are examples of two-dimensional fractal structures. Here we will consider more complex examples of three-dimensional fractals.
The fractal can be clearly imagined (describe), as a geometric shape or body (having in mind that both there are many, in this case, a set of points), the details of which have the same shape as the original figure itself. That is, it is a self-like structure, considering the details of which with an increase, we will see the same form as without increasing. Whereas in the case of a conventional geometric shape (not fractal), with an increase we will see parts that have a simpler form than the original figure itself. For example, with a sufficiently large increase, the part of the ellipse looks like a straight line. With fractals, this does not happen: with any increases, we will again see the same complex shape, which will repeat again and again.
Benoit Mandelbrot (Benoit Mandelbrot), the founder of the science of fractals, in his article Fractals and art in the name of science wrote: "Fractals are geometric forms that are equally complex in their details, as in its general form. That is, if Part of the fractal will be increased to the size of the whole, it will look like an integer, or exactly, or, possibly, with a small deformation. "
The matrix A -1 is called the inverse matrix in relation to the matrix A, if A * A -1 \u003d E, where e is a single matrix N-order. Reverse matrix may exist only for square matrices.
Appointment of service. With this service in online mode, you can find algebraic supplements, a transposed matrix A T, the Allied matrix and a reverse matrix. The decision is carried out directly on the site (in online mode) and is free. The results of the calculations are made in the Word report and in Excel format (i.e., it is possible to check the solution). See Example of registration.
Instruction. To obtain a solution, you must specify the dimension of the matrix. Next, in the new dialog box, fill out the matrix a.
See also Inverse Matrix by Jordan-Gauss
Algorithm for the return matrix
- Finding a transposed matrix a t.
- Definition of algebraic additions. Replace each element of the matrix by its algebraic addition.
- The preparation of the return matrix from algebraic additions: each element of the resulting matrix is \u200b\u200bdivided into the determinant of the original matrix. The resulting matrix is \u200b\u200breverse for the original matrix.
- Determine whether square matrix. If not, the reverse matrix does not exist for it.
- Calculation of the determinant of the matrix a. If it is not equal to zero, we continue the solution, otherwise there is no reverse matrix.
- Definition of algebraic additions.
- Filling the union (mutual attached) matrix C.
- Drawing up a reverse matrix of algebraic additions: Each element of the attached matrix C is divided into the determinant of the original matrix. The resulting matrix is \u200b\u200breverse for the original matrix.
- Check: Move the original and obtained matrix. As a result, a single matrix should be obtained.
Example number 1. We write the matrix in the form:
| A -1 \u003d. |
|
Another algorithm for finding a reverse matrix
We give another diagram of finding the return matrix.- We find the determinant of this square matrix a.
- We find algebraic additions to all elements of the matrix a.
- Record algebraic supplements of elements of rows in columns (transposition).
- We divide each element of the resulting matrix to the determinant of the matrix a.
A special case: Reverse, with respect to a single matrix E, is a single matrix E.