Matrices and determinants were discovered in the eighteenth and nineteenth centuries. Initially, their development concerned the transformation of geometric objects and the solution of systems of linear equations. Historically, an early focus was on the determinant. In modern linear algebra processing methods, matrices are considered first. It’s worth a little thought on this issue.
Answers given by this field of knowledge
Matrices provide a theoretically and practically useful way to solve many problems, such as:
- systems of linear equations;
- balance of solids (in physics);
- graph theory;
- Leontief model of economics;
- forestry;
- computer graphics and tomography;
- genetics;
- cryptography;
- Electricity of the net;
- fractal.
In fact, the matrix algebra for dummies has a simplified definition. It is expressed as follows: this is a scientific field of knowledge in which the values considered are studied, analyzed and investigated in full. In this section of algebra, various operations on the matrices under study are studied.
How to work with matrices
These values are considered equal if they have the same dimensions and each element of one is equal to the corresponding element of the other. It is possible to multiply the matrix by any constant. This given is called scalar multiplication. Example: 2 = [1234] = [2⋅12⋅32⋅22⋅4] = [2468].
Matrices of the same size can be added and subtracted by inputs, and values of compatible sizes can be multiplied. Example: add two A and B: A = [21−10] B = [1423]. This is possible, since A and B - both matrices have two rows and the same number of columns. It is necessary to add each element in A to the corresponding element in B: A + B = [2 + 11 + 2−1 + 40 + 3] = [3333]. Similarly, the matrix is subtracted in the algebra.
Matrix multiplication is a little different. Moreover, there can be many cases and options, as well as solutions. If we multiply the matrix Ap * q and Bm * n, then the product Ap × q + Bm × n = [AB] p × n. The element in the gth row and the hth column AB is the sum of the product of the corresponding elements in g A and h B. It is only possible to multiply two matrices if the number of columns in the first and rows in the second are equal. Example: to satisfy the condition for the considered A and B: A = [1−130] B = [2−11214]. This is possible because the first matrix contains 2 columns, and the second contains 2 rows. AB = [1⋅2 + 3⋅ − 1−1⋅2 + 0⋅ − 11⋅1 + 3⋅2−1⋅1 + 0⋅21⋅1 + 3⋅4−1⋅1 + 0⋅4] = [−1−27−113−1].
Basic Matrix Information
The values in question organize information, such as variables and constants, and store them in rows and columns, they are usually called C. Each position in the matrix is called an element. Example: C = [1234]. Consists of two rows and two columns. Element 4 is in row 2 and column 2. You can usually name a matrix after its size, one with the name Cm * k has m rows and k columns.
Extended matrices
The meanings considered are incredibly useful things that arise in many different application areas. Matrices were originally based on systems of linear equations. Given the following structure of inequalities, it is necessary to take into account the following related augmented matrix:
2x + 3y - z = 6
–X - y - z = 9
x + y + 6z = 0.
Write down the coefficients and values of the answers, including all the minus signs. If the element is with a negative number, then it will be equal to "1". That is, given the system of (linear) equations, it is possible to associate a matrix with it (a grid of numbers inside brackets). It is the one that contains only the coefficients of the linear system. This is called an "expanded matrix." The grid containing the coefficients from the left side of each equation was “supplemented” by the answers from the right side of each equation.
Records, that is, the values of the B matrix correspond to the values of x-, y- and z in the original system. If it is correctly arranged, then it is checked first of all. Sometimes you need to rearrange the terms or insert zeros as holders of places in the studied or studied matrix.
Given the following system of equations, you can immediately write a related augmented matrix:
x + y = 0
y + z = 3
z - x = 2.
First you need to rearrange the system as:
x + y = 0
y + z = 3
–X + z = 2.
Then it is possible to write a related matrix as: [11000113-1012]. When forming an extended one, use zero for any record where the corresponding spot in the system of linear equations is empty.
Matrix Algebra: Operation Properties
If it is necessary to form elements only from coefficient values, then the considered value will look like this: [110011-101]. This is called a "matrix of coefficients."
Given the following extended matrix algebra, it is necessary to improve it and add a connected linear system. At the same time, it is important to remember that they require that the variables are well and neatly arranged. And usually when there are three variables, use x, y and z in that order. Therefore, a connected linear system should be:
x + 3y = 4
2y - z = 5
3x + z = -2.
Matrix size
The items in question are often referred to by their indicators. The size of the matrix in algebra is given in the form of a measurement, since a room can be called differently. Measured values are rows and columns, not width and length. For example, matrix A:
[1234]
[2345]
[3456].
Since A has three rows and four columns, the size of A is 3 × 4.
→
↓
Lines go to the side. Columns go up and down. “Row” and “column” are specifications and are not interchangeable. Matrix sizes are always specified with the number of rows and then the number of columns. Following this convention, the following B:
[123]
[234] is 2 × 3. If a matrix has the same number of rows as columns, then it is called “square”. For example, the values of the coefficients above:
[110]
[011]
[-101] is a 3 × 3 square matrix.
Matrix Conventions and Formatting
Note regarding formatting: for example, when you need to write a matrix, it is important to use brackets []. Bars of absolute value || are not used, since in this context they have a different direction. In no case shall parentheses or braces {} be used. Or some other grouping symbol or none at all, as these presentations have no meaning. In algebra, a matrix is always inside square brackets. Only the correct notation must be used, or the answers received may be considered distorted.
As mentioned earlier, the values contained in the matrix are called records. For some reason, the items in question are usually written in capital letters, such as A or B, and entries are indicated using the corresponding lowercase, but with indexes. In matrix A, values are usually called “ai, j”, where i is row A and j is column A. For example, a3,2 = 8. The entry a1,3 is 3.
For smaller matrices, those with less than ten rows and columns, the comma in the subscript is sometimes omitted. For example, “a1,3 = 3” can be written as “a13 = 3”. Obviously, this will not work for large matrices, since a213 will be obscure.
Matrix Types
Sometimes classified according to their record configurations. For example, such a matrix, which has all zero entries below the diagonal from top to left, down, and right, the “diagonal”, is called the upper triangular. Among other things, there may be other types and types, but they are not very useful. As a rule, they are generally perceived as upper triangular. Values with non-zero indicators only horizontally are called diagonal. Similar types have nonzero entries in which all 1, such answers are called identical (for reasons that will become clear when it is learned and it is clear how to multiply the values in question). There are many similar studied indicators. The identity 3 × 3 is denoted by I3. Similarly, a 4x4 identity equals I4.
Matrix Algebra and Linear Spaces
It should be noted that the triangular matrices are square. But the diagonals are triangular. In view of this, they are square. And identities are considered diagonals and, therefore, triangular and square. When it is required to describe a matrix, usually its own most definite classification is simply indicated, since this implies all the others. The following research options can be classified: [[9 10 11 12] [5 6 7 8] [1 2 3 4]] can be as 3 × 4. In this case, they are not square. Therefore, the values cannot be any else. The following classification: [[9 0 4] [3 -2 3] [1 6 7]] it is possible as 3 × 3. But at the same time it is considered square, and there is nothing special about it. The classification of the following data: [[0 8 -4] [1 0 2] [0 0 5]], as 3 × 3 is the upper triangular, but it is not diagonal. True, in the considered values there may be additional zeros on the located and indicated space or above it. The classification under study is further: [[0 0 1] [1 0 0] [0 1 0]], where it is presented as a diagonal and, moreover, the entries are all 1. Then this is the identity 3 × 3, I3.
Since similar matrices are by definition square, you only need to use one index to find their sizes. In order for the two matrices to be equal, they must be of the same parameter, and also have the same records in the same places. For example, suppose there are two of the following elements under consideration: A = [[1 3 0] [-2 0 0]] and B = [[1 3] [-2 0]]. These values may not be the same because they are different in size.
Even if A and B are: A = [[3 6] [2 5] [1 4]] and B = [[1 2 3] [4 5 6]] - they are still not the same. A and B have six records each, and also have the same numbers, but this is not enough for matrices. A is 3 × 2. A B is a 2 × 3 matrix. And for 3 × 2 it is not 2 × 3. It does not matter if A and B have the same amount of data or even the same numbers as the records. If A and B do not have the same size and shape, but have identical values in similar places, they are not equal.
Similar operations in this area
This property of matrix equality can be turned into tasks for independent research. For example, two matrices are given, and it is indicated that they are equal. In this case, it will be necessary to use this equality to study and obtain answers to the values of variables.
Examples and solutions of matrices in algebra can be varied, especially when it comes to equalities. Given that the following matrices are considered, it is necessary to find the values of x and y. In order for A and B to be equal, they must have the same size and shape. In fact, they are such, because each of them is 2 × 2 matrices. And they should have the same meaning in the same places. Then a1,1 should be equal to b1,1, a1,2 should be equal to b1,2, etc. The entries a1,2 and a2,1 are clearly equal to the elements b1,2 and b2,1, respectively (by checking, i.e. just looking them). But, a1,1 = 1, obviously, is not equal to b1,1 = x. For A identical to B, the record must have a1,1 = b1,1, so it can equal 1 = x. Similarly, the indices a2,2 = b2,2, so 4 = y. Then the solution is: x = 1, y = 4. Given that the following matrices are equal, you need to find the values of x, y and z. To have A = B, the coefficients must have all entries equal. That is, a1,1 = b1,1, a1,2 = b1,2, a2,1 = b2,1 and so on. In particular, should:
4 = x
-2 = y + 4
3 = z / 3.
As can be seen from the selected matrices: with 1,1-, 2,2- and 3,1-elements. Solving these three equations, we get the answer: x = 4, y = -6 and z = 9. The matrix algebra and matrix operations are different from what everyone is used to, but they are not reproducible.
Additional information in this area
Linear matrix algebra is the study of such sets of equations and their transformation properties. This area of knowledge allows one to analyze rotations in space, approximate the least squares, solve related differential equations, determine a circle passing through three given points, and also solve many other questions of mathematics, physics and technology. The linear matrix algebra is not really the technical meaning of the word used, that is, the vector space v over the field f, etc.
The matrix and determinant are extremely useful tools in linear algebra. One of the central tasks is to solve the matrix equation A x = b, for x. Although this can theoretically be solved using the inverse x = A -1 b. Other methods, such as Gaussian exception, are numerically more reliable.
In addition to being used to describe the study of linear sets of equations, the above term is also used to describe a certain type of algebra. In particular, L over the field F has a ring structure with all the usual axioms for internal addition and multiplication together with distribution laws. Therefore, it gives it more structure than a ring. A linear matrix algebra also admits an external operation of multiplication by scalars that are elements of the underlying field F. For example, the set of all considered transformations from the vector space V to itself over the field F is formed over F. Another example of linear algebra is the set of all real square matrices over the field R real numbers.