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, the early emphasis was on the determinant. In modern linear algebra processing methods, matrices are considered first. It is worth pondering this question for a while.
Answers from this area of knowledge
Matrices provide a theoretically and practically useful way to solve many problems, such as:
- systems of linear equations;
- equilibrium of solids (in physics);
- graph theory;
- Leontief's economic model;
- forestry;
- computer graphics and tomography;
- genetics;
- cryptography;
- electric networks;
- fractal.
In fact, matrix algebra for "dummies" has a simplified definition. It is expressed as follows: this is a scientific field of knowledge in whichthe values in question are studied, analyzed and fully explored. 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 a 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 because A and B are both matrices with 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]. Matrices are subtracted in the same way in algebra.
Matrix multiplication works a little differently. Moreover, there can be many cases and options, as well as solutions. If we multiply the matrix Apq and Bmn, then the product Ap×q+Bm×n=[AB]p×n. The entry in the gth row and the hth column of AB is the sum of the product of the corresponding entries 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: fulfill the condition for 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 information about matrices
The values in question organize information such as variables and constants and store them in rows and columns, 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 dimensions, the one named Cmk has m rows and k columns.
Expanded matrices
Considerations are incredibly useful things that come up in many different application areas. Matrices were originally based on systems of linear equations. Given the following structure of inequalities, the following complemented matrix needs to be taken into account:
2x + 3y – z=6
–x – y – z=9
x + y + 6z=0.
Write down coefficients and answer values, including all minus signs. If the element with a negative number, then it will be equal to "1". That is, given a system of (linear) equations, it is possible to associate a matrix (grid of numbers inside brackets) with it. It is the one that contains only the coefficients of the linear system. This is called the "expanded matrix". The grid containing the coefficients from the left side of each equation has been "padded" with the answers from the right side of each equation.
Records, that isthe B values of the matrix correspond to the x-, y-, and z values in the original system. If it is properly arranged, then first of all check it. Sometimes you need to rearrange the terms or insert zeros as holders of places in the matrix being studied or studied.
Given the following system of equations, we can immediately write the associated augmented matrix:
x + y=0
y + z=3
z – x=2.
First, be sure to rearrange the system as:
x + y=0
y + z=3
–x + z=2.
Then it is possible to write the associated matrix as: [11000113-1012]. When forming an extended one, it is worth using zero for any record where the corresponding spot in the system of linear equations is empty.
Matrix Algebra: Properties of Operations
If it is necessary to form elements only from coefficient values, then the considered value will look like this: [110011-101]. This is called the "coefficient matrix".
Taking into account the following extended matrix algebra, it is necessary to improve it and add the associated linear system. That being said, it's important to remember that they require variables to be well-arranged and neat. And usually when there are three variables, use x, y and z in that order. Therefore, the associated 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 performance. The size of a matrix in algebra is given asmeasurements, since the room can be called differently. Measured measures of 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 sideways. The 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 a "square". For example, coefficient values from above:
[110]
[011]
[-101] is a 3×3 square matrix.
Matrix notation and formatting
Formatting note: For example, when writing a matrix, it is important to use brackets. Absolute value bars || are not used because they have a different direction in this context. Parentheses or curly braces {} are never used. Or some other grouping symbol, or none at all, as these presentations don't have any meaning. In algebra, a matrix is always inside square brackets. Only correct notation must be used, or responses may be considered garbled.
As mentioned earlier, the values contained in a matrix are called records. For whatever reason, the elements in question are usually writtencapital letters, such as A or B, and entries are specified using the corresponding lowercase letters, but with subscripts. In matrix A, the values are usually called "ai, j", where i is the row of A and j is the column of A. For example, a3, 2=8. The entry for a1, 3 is 3.
For smaller matrices, those with fewer than ten rows and columns, the subscript comma is sometimes omitted. For example, "a1, 3=3" could be written as "a13=3". Obviously this won't work for large matrices as a213 will be obscure.
Matrix types
Sometimes classified according to their record configurations. For example, such a matrix that has all zero entries below the diagonal top-left-bottom-right "diagonal" is called upper triangular. Among other things, there may be other kinds and types, but they are not very useful. Generally, mostly perceived as upper triangular. Values with non-zero exponents only horizontally are called diagonal values. Similar types have non-zero entries in which all are 1, such answers are called identical (for reasons that will become clear when it is learned and understood how to multiply the values in question). There are many similar research indicators. The 3 × 3 identity is denoted by I3. Similarly, the 4 × 4 identity is I4.
Matrix Algebra and Linear Spaces
Note that triangular matrices are square. But the diagonals are triangular. In view of this, they aresquare. And identities are considered diagonals and, therefore, triangular and square. When it is required to describe a matrix, one usually simply specifies one's own most specific classification, since this implies all the others. The following research options can be classified:as 3 × 4. In this case, they are not square. Therefore, the values cannot be anything else. The following classification:is possible as 3 × 3. But it is considered a square, and there is nothing special about it. Classification of the following data:as 3 × 3 upper triangular, but it is not diagonal. True, in the values under consideration there may be additional zeros on or above the located and indicated space. The classification under study is further: [0 0 1] [1 0 0] [0 1 0], where it is represented as a diagonal and, moreover, the entries are all 1. Then this is a 3 × 3 identity, I3.
Since analogous matrices are by definition square, you only need to use a single index to find their dimensions. For two matrices to be equal, they must have the same parameter and have the same entries in the same places. For example, suppose there are two elements under consideration: A=[1 3 0] [-2 0 0] and B=[1 3] [-2 0]. These values cannot be the same as 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 same thing. A and B each havesix entries and also have the same numbers, but this is not enough for matrices. A is 3×2. And B is a 2×3 matrix. A for 3×2 is not 2×3. It doesn't matter if A and B have the same amount of data or even the same numbers as the records. If A and B are not the same size and shape, but have identical values in similar places, they are not equal.
Similar operations in the area under consideration
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, you will need to use this equality to explore and get answers for the values of the 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 x and y values. For A and B to be equal, they must be the same size and shape. In fact, they are such, because each of them is 2 × 2 matrices. And they should have the same values in the same places. Then a1, 1 must equal b1, 1, a1, 2 must equal b1, 2, and so on. them). But, a1, 1=1 is obviously not equal to b1, 1=x. For A to be identical to B, the entry must have a1, 1=b1, 1, so it is capable of being 1=x. Similarly, the indices a2, 2=b2, 2, so 4=y. Then the solution is: x=1, y=4. Given that the followingmatrices 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, must:
4=x
-2=y + 4
3=z / 3.
As you can see 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. 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 similar sets of equations and their transformation properties. This field of knowledge allows you to analyze rotations in space, approximate least squares, solve associated differential equations, determine a circle passing through three given points, and solve many other problems in mathematics, physics and technology. The linear algebra of a matrix is not really the technical sense of the word used, that is, a vector space v over a field f, etc.
Matrix and determinant are extremely useful linear algebra tools. One of the central tasks is the solution of the matrix equation Ax=b, for x. Although this could theoretically be solved using the inverse x=A-1 b. Other methods, such as Gaussian elimination, are numerically more robust.
In addition to being used to describe the study of linear sets of equations, the specifiedthe above term is also used to describe a particular type of algebra. In particular, L over a field F has the structure of a ring with all the usual axioms for internal addition and multiplication, together with distributive laws. Therefore, it gives it more structure than a ring. Linear matrix algebra also admits the outer 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.
