EntranceSpecial types of matrices. Types of matrices
Definition 1.Size matrix call a table of numbers
consisting of a row of columns 
In this case, the numbers 1 are called matrix elements The matrix is called square matrix of dimension if the number of its rows matches the number of columns
Often a matrix is denoted as follows: If we want to indicate the dimensions of the matrix, we will write and call the matrix itself a matrix.
Actions addition and subtraction over matrices same size are defined by the equalities:
(i.e., when adding or subtracting matrices, their elements located in the same places are added (subtracted accordingly).
Multiplying a matrix by a number is determined by equality
(i.e. when multiplying a matrix by a number, each element of this matrix must be multiplied by this number).
Matrices can be multiplied by each other only if they are sizes agreed , i.e., when the number of columns of the first matrix is equal to the number of rows of the second matrix:
First, determine the product of the row vector by
column vector (having the same number of components):
Then determine
V) product of matrices with consistent dimensions is called a matrix element of which is obtained by multiplying a row of the matrix by a column of the matrix
For example, 
The following special types of matrices are often encountered:
1. Identity matrix:
2. Diagonal
matrix: 
(here and in the matrix, all elements outside the main diagonal are equal to zero).
3.Triangular matrix: 
4. Trapezoidal matrix: 
When solving linear systems of equations, you will encounter matrices stepped type. To describe them, we introduce the concept line support element. This the non-null first element of the string from the left. For example, in the line the element (-5) is the support element (here and below the support element is indicated in the frame).
Definition 2. The matrix is called a matrix stepped type, if it contains:
a) the supporting element of each row is located to the right the supporting element of the previous row;
b) If a matrix has a zero row, then all its subsequent rows are also zero.
It is clear that diagonal, upper triangular and trapezoidal matrices are step matrices. Another example of a stepped matrix:
2. Matrix determinants and their properties
We have already dealt with second- and third-order determinants in previous lectures. Let us now give the general concept of the determinant of order by induction. Any square matrix of the form
matches the number
defined below (see Definition 5) and called determinant (or determinant) of the matrix Now let's introduce the concept minor matrices.
Definition 3. In a matrix, at the intersection of any rows and columns there is an order matrix. Matrix determinant is called the th order minor of the matrix
Clearly, there may be several such minors. Let now the matrix be square.
Definition 4. The minor order obtained from a matrix after deleting its row and column, is called the complementary minor of the element this matrix (designation:). The number is called algebraic complement of an element matrices.
Definition 5. Let an arbitrary row is highlighted in a square matrix Matrix determinant called number
(i.e. the sum of the products of the elements of the th row and their algebraic complements). Often the determinant of a matrix is denoted as follows:
As we noted above, the determinant of order is calculated by induction: if the rule for calculating determinants of order is known, then the determinant of order is calculated using formula (1). Previously, the rules for calculating determinants of the second and third orders were given, so using formula (1) you can calculate determinants of the fourth order and higher. For example,
Let's list the main properties of determinants. First, note that a matrix obtained from a matrix by replacing rows with columns with the same numbers is called trans-
sponsored to the matrix. Designation:
1) When transposing a matrix, its determinant does not change:
2) When any two rows (or two columns) of a matrix are rearranged, its determinant changes sign to the opposite one.
3) A determinant that has a zero row (or a zero column) is equal to zero.
4) The determinant in which the elements of one row (or column) are proportional to the elements of another row (or column) is equal to zero.
5) The common factor of the elements of any row (or column) can be taken beyond the determinant sign: 
6) If you add another line to any line of the determinant, multiplied by any number, the determinant will not change. The same is true for determinant columns.
7) (sum of determinants) 
8) The determinant of the product of two square matrices of the same dimension is equal to the product of the determinants of these matrices:
Proof all these properties are carried out using Definition 5. Let us prove, for example, Property 5. We have
Property 5 has been proven.
Ticket 17:
Question 1: Definition of a parabola. Derivation of the equation:
Definition. A parabola is a set of points on a plane, each of which is at the same distance from a given point, called the focus, and from a given straight line, called the directrix and not passing through the focus.
Let's place the origin of coordinates in the middle between the focus and the directrix.
The value p (the distance from the focus to the directrix) is called the parameter of the parabola. Let us derive the canonical equation of the parabola.
From geometric relationships: AM = MF; AM = x + p/2;
MF2 = y2 + (x – p/2)2
(x + p/2)2 = y2 + (x – p/2)2
x2 +xp + p2/4 = y2 + x2 – xp + p2/4
Directrix equation: x = -p/2.
Question 2: Cauchy's Theorem:
Theorem: Let the functions and be differentiable on the interval and continuous for and , and for all . Then in the interval there is a point such that
Geometric meaning : The data of the theorem is that inside there is a point t 0, the angular coefficients at which are calculated by the equality:
Proof.
Let us first prove that 
, that is, that the fraction on the left side of the formula makes sense. Indeed, for this difference we can write the formula for finite increments:
at some . But on the right side of this formula both factors are non-zero.
To prove the theorem, we introduce an auxiliary function
The function is obviously differentiable for all and continuous at the points and , since the functions and have these properties. Moreover, it is obvious that when it turns out . Let us show that and:
This means that the function satisfies the conditions of Rolle’s theorem on the segment. Therefore, there is such a point that.
Let us now calculate the derivative of the function:
We get that
from which we obtain the statement of the theorem:
Comment: We can consider functions and coordinates of a point moving on a plane, which describes a line connecting the starting point with the end point. (Then the equations and parametrically define a certain dependence, the graph of which is the line.)
Fig. 5.6. A chord is parallel to some tangent to the curve
The ratio, as is easy to see from the drawing, then sets the angular coefficient of the chord connecting the points and. At the same time, according to the formula for the derivative of a function specified parametrically, we have: 
. This means that a fraction is the angular coefficient of the tangent to the line at some point 
. Thus, the statement of the theorem means, from a geometric point of view, that there is a point on the line such that the tangent drawn at this point is parallel to the chord connecting the extreme points of the line. But this is the same statement that constituted the geometric meaning of Lagrange's theorem. Only in Lagrange's theorem was the line specified by an explicit dependence, and in Cauchy's theorem by a dependence specified in parametric form.
Ticket 18:
Question 1: The concept of a matrix. Matrix classification:
Definition.
A matrix of size mn, where m is the number of rows, n is the number of columns, is a table of numbers arranged in a certain order. These numbers are called matrix elements. The location of each element is uniquely determined by the number of the row and column at the intersection of which it is located. The elements of the matrix are denoted by aij, where i is the row number and j is the column number. A = 
Classification of matrices:.
A matrix can consist of either one row or one column. Generally speaking, a matrix can even consist of one element.
Definition . If the number of matrix columns is equal to the number of rows (m=n), then the matrix is called square.
Definition
. View matrix: 
= E, is called the identity matrix.
Definition. If amn = anm, then the matrix is called symmetric. Example. - symmetric matrix
Definition
. Square matrix of the form 
called diagonal matrix
.
Question 2: Lagrange's theorem:
Theorem:
Let the function be differentiable on the interval and continuous at the points and . Then there will be a point such that 
Geometric meaning:
Let us first give a geometric illustration of the theorem. Let's connect the end points of the graph on a segment with a chord. Final increments and 
- these are the sizes of the legs of a triangle, the hypotenuse of which is the drawn chord.
Fig. 5.5. The tangent at some point is parallel to the chord
The ratio of the final increments and is the tangent of the angle of inclination of the chord. The theorem states that a tangent can be drawn to the graph of a differentiable function at some point, which will be parallel to the chord, that is, the angle of inclination of the tangent () will be equal to the angle of inclination of the chord (). But the presence of such a tangent is geometrically obvious.
Note that the drawn chord connecting the points and is the graph of a linear function. Since the slope of this linear function is obviously equal to 
, That 
Proof of Lagrange's theorem. Let us reduce the proof to the application of Rolle's theorem. To do this, we introduce an auxiliary function, that is
notice, that 
and (by constructing the function ). Since a linear function is differentiable for all , the function satisfies, therefore, all the properties listed in the conditions of Rolle’s theorem. Therefore, there is such a point that By philosophy: answers to exam papers Cheat sheet >> Philosophy
Crib By philosophy: answers to exam papers... painting, sculpture and architecture, work By mathematics, biology, geology, anatomy are dedicated to man... self-discipline, orient oneself towards higher goals. Basic thoughts of ancient Eastern...
Crib By Logic: Answers to exam papers
Cheat sheet >> PhilosophyValery Vechkanov Crib By logic Vladimir Eduardovich Vechkanov Crib By logic: ... human thinking. Physiology highest nervous activity reveals naturally... propositional function is widely used in mathematics. All equations with one...
Crib By Econometrics (1)
Cheat sheet >> EconomicsStatistics; economic statistics; highest mathematicians. Means. contribution to development... By degree of crowding, By direction and By analytical alignment. By... change in opposite directions. By analytical alignment: - linear connections...
Definition 1.Matrix, dimensions
p 
q
is a rectangular table of numbers containing p lines and q columns. The numbers that make up the matrix are called elements of the matrix.
Formulas (3.1) involve matrices of dimensions k
n And n
k.
Rules for writing matrices:
1). Matrices are denoted in capital Latin letters. The subscript of the letter is the dimension of the matrix. This index can be omitted if the dimension of the matrix is known.
2). The symbol “T”, as before (see page), means transposition , that is, replacing rows with columns.
3). The table of numbers itself is written in parentheses.
4). If the matrix is not transposed, then the first index of its element means the row number, the second - the column number.
Matrices are very widely used in mathematics, computer science, economics, etc. The rules for operating matrices are one of the questions of linear algebra. It was from linear algebra that the concept of a matrix came to other areas of knowledge.
Classification of matrices by dimension.
|
Matrix name |
Matrix column |
Matrix row |
Rectangular matrix |
Square matrix |
|
Matrix type |
|
|
|
|
|
|
|
|
Square matrices play a special role in linear algebra. Let's look at them in more detail. First of all, we note that in case of a square matrix they are not talking about the dimension of the matrix 
, and about
matrix order, equal
n.
A square matrix has main and secondary diagonals :
secondary diagonal main diagonal
Classification of square matrices
|
Name |
Identity matrix |
Diagonal matrix |
Triangular matrices |
|
|
upper triangular |
inferior triangular |
|||
|
Matrix type |
|
|
|
|
|
Fourth order identity matrix
|
Fourth order diagonal matrix
|
Upper triangular matrix of third order
|
Fifth order lower triangular matrix
|
|
Considering a matrix as a system of arithmetic vectors, it is easy to understand what equality of matrices means and how linear operations are performed on them.
Matrix equality
Two matrices of the same dimension are equal each other if their corresponding elements 7 are equal.
For convenience of writing statements regarding matrices, the following symbolism is often used:
|
|
In such an entry the symbol “ 
» – designation of any of the matrix elements, indices i
,
j
- current
indices
, variables, running values, . The expanded formula (3.2) looks like this:
|
|
=
A matrix is a special object in mathematics. It is depicted in the form of a rectangular or square table, composed of a certain number of rows and columns. In mathematics there is a wide variety of types of matrices, varying in size or content. The numbers of its rows and columns are called orders. These objects are used in mathematics to organize the recording of systems of linear equations and conveniently search for their results. Equations using a matrix are solved using the method of Carl Gauss, Gabriel Cramer, minors and algebraic additions, as well as many other methods. The basic skill when working with matrices is reduction to However, first, let's figure out what types of matrices are distinguished by mathematicians.
Null type
All components of this type of matrix are zeros. Meanwhile, the number of its rows and columns is completely different.
Square type
The number of columns and rows of this type of matrix is the same. In other words, it is a “square” shaped table. The number of its columns (or rows) is called the order. Special cases are considered to be the existence of a second-order matrix (2x2 matrix), fourth-order (4x4), tenth-order (10x10), seventeenth-order (17x17) and so on.
Column vector
This is one of the simplest types of matrices, containing only one column, which includes three numerical values. It represents a number of free terms (numbers independent of variables) in systems of linear equations.
View similar to the previous one. Consists of three numerical elements, in turn organized into one line.
Diagonal type
Numerical values in the diagonal form of the matrix take only the components of the main diagonal (highlighted in green). The main diagonal begins with the element located in the upper left corner and ends with the element in the lower right, respectively. The remaining components are equal to zero. The diagonal type is only a square matrix of some order. Among the diagonal matrices, one can distinguish the scalar one. All its components take the same values.
A subtype of diagonal matrix. All its numerical values are units. Using a single type of matrix table, one performs its basic transformations or finds a matrix inverse to the original one.
Canonical type
The canonical form of the matrix is considered one of the main ones; casting to it is often necessary for work. The number of rows and columns in a canonical matrix varies, and it does not necessarily belong to the square type. It is somewhat similar to the identity matrix, but in its case not all components of the main diagonal take on a value equal to one. There can be two or four main diagonal units (it all depends on the length and width of the matrix). Or there may be no units at all (then it is considered zero). The remaining components of the canonical type, as well as the diagonal and unit elements, are equal to zero.
Triangular type
One of the most important types of matrix, used when searching for its determinant and when performing simple operations. The triangular type comes from the diagonal type, so the matrix is also square. The triangular type of matrix is divided into upper triangular and lower triangular.
In an upper triangular matrix (Fig. 1), only elements that are above the main diagonal take a value equal to zero. The components of the diagonal itself and the part of the matrix located under it contain numerical values.
In the lower triangular matrix (Fig. 2), on the contrary, the elements located in the lower part of the matrix are equal to zero.
The type is necessary to find the rank of a matrix, as well as for elementary operations on them (along with the triangular type). The step matrix is so named because it contains characteristic "steps" of zeros (as shown in the figure). In the step type, a diagonal of zeros is formed (not necessarily the main one), and all elements under this diagonal also have values equal to zero. A prerequisite is the following: if there is a zero row in the step matrix, then the remaining rows below it also do not contain numerical values.
Thus, we examined the most important types of matrices necessary to work with them. Now let's look at the problem of converting the matrix into the required form.
Reducing to triangular form
How to bring a matrix to a triangular form? Most often in tasks you need to transform a matrix into a triangular form in order to find its determinant, otherwise called a determinant. When performing this procedure, it is extremely important to “preserve” the main diagonal of the matrix, because the determinant of a triangular matrix is equal to the product of the components of its main diagonal. Let me also recall alternative methods for finding the determinant. The determinant of the square type is found using special formulas. For example, you can use the triangle method. For other matrices, the method of decomposition by row, column or their elements is used. You can also use the method of minors and algebraic matrix additions.
Let us analyze in detail the process of reducing a matrix to a triangular form using examples of some tasks.
Exercise 1
It is necessary to find the determinant of the presented matrix using the method of reducing it to triangular form.
The matrix given to us is a third-order square matrix. Therefore, to transform it into a triangular shape, we will need to zero out two components of the first column and one component of the second.
To bring it to triangular form, we start the transformation from the lower left corner of the matrix - from the number 6. To turn it to zero, multiply the first row by three and subtract it from the last row.
Important! The top row does not change, but remains the same as in the original matrix. There is no need to write a string four times larger than the original one. But the values of the strings whose components need to be set to zero are constantly changing.
Only the last value remains - the element of the third row of the second column. This is the number (-1). To turn it to zero, subtract the second from the first line.
Let's check:
detA = 2 x (-1) x 11 = -22.
This means that the answer to the task is -22.
Task 2
It is necessary to find the determinant of the matrix by reducing it to triangular form.
The presented matrix belongs to the square type and is a fourth-order matrix. This means that it is necessary to turn three components of the first column, two components of the second column and one component of the third to zero.
Let's start reducing it with the element located in the lower left corner - with the number 4. We need to turn this number to zero. The easiest way to do this is to multiply the top line by four and then subtract it from the fourth. Let's write down the result of the first stage of transformation.
So the fourth row component is set to zero. Let's move on to the first element of the third line, to the number 3. We perform a similar operation. We multiply the first line by three, subtract it from the third line and write down the result.
We managed to turn to zero all the components of the first column of this square matrix, with the exception of the number 1 - an element of the main diagonal that does not require transformation. Now it is important to preserve the resulting zeros, so we will perform the transformations with rows, not with columns. Let's move on to the second column of the presented matrix.
Let's start again at the bottom - with the element of the second column of the last row. This number is (-7). However, in this case it is more convenient to start with the number (-1) - the element of the second column of the third row. To turn it to zero, subtract the second from the third line. Then we multiply the second line by seven and subtract it from the fourth. We got zero instead of the element located in the fourth row of the second column. Now let's move on to the third column.
In this column, we need to turn only one number to zero - 4. This is not difficult to do: we simply add a third to the last line and see the zero we need.
After all the transformations made, we brought the proposed matrix to a triangular form. Now, to find its determinant, you only need to multiply the resulting elements of the main diagonal. We get: detA = 1 x (-1) x (-4) x 40 = 160. Therefore, the solution is 160.
So, now the question of reducing the matrix to triangular form will not bother you.
Reducing to a stepped form
For elementary operations on matrices, the stepped form is less “in demand” than the triangular one. It is most often used to find the rank of a matrix (i.e., the number of its non-zero rows) or to determine linearly dependent and independent rows. However, the stepped type of matrix is more universal, as it is suitable not only for the square type, but also for all others.
To reduce a matrix to stepwise form, you first need to find its determinant. The above methods are suitable for this. The purpose of finding the determinant is to find out whether it can be converted into a step matrix. If the determinant is greater or less than zero, then you can safely proceed to the task. If it is equal to zero, it will not be possible to reduce the matrix to a stepwise form. In this case, you need to check whether there are any errors in the recording or in the matrix transformations. If there are no such inaccuracies, the task cannot be solved.
Let's look at how to reduce a matrix to a stepwise form using examples of several tasks.
Exercise 1. Find the rank of the given matrix table.
Before us is a third-order square matrix (3x3). We know that to find the rank it is necessary to reduce it to a stepwise form. Therefore, first we need to find the determinant of the matrix. Let's use the triangle method: detA = (1 x 5 x 0) + (2 x 1 x 2) + (6 x 3 x 4) - (1 x 1 x 4) - (2 x 3 x 0) - (6 x 5 x 2) = 12.
Determinant = 12. It is greater than zero, which means that the matrix can be reduced to a stepwise form. Let's start transforming it.
Let's start it with the element of the left column of the third line - the number 2. Multiply the top line by two and subtract it from the third. Thanks to this operation, both the element we need and the number 4 - the element of the second column of the third row - turned to zero.
We see that as a result of the reduction, a triangular matrix was formed. In our case, we cannot continue the transformation, since the remaining components cannot be reduced to zero.
This means that we conclude that the number of rows containing numerical values in this matrix (or its rank) is 3. The answer to the task: 3.
Task 2. Determine the number of linearly independent rows of this matrix.
We need to find strings that cannot be converted to zero by any transformation. In fact, we need to find the number of non-zero rows, or the rank of the presented matrix. To do this, let us simplify it.
We see a matrix that does not belong to the square type. It measures 3x4. Let's also start the reduction with the element of the lower left corner - the number (-1).
Its further transformations are impossible. This means that we conclude that the number of linearly independent lines in it and the answer to the task is 3.
Now reducing the matrix to a stepped form is not an impossible task for you.
Using examples of these tasks, we examined the reduction of a matrix to a triangular form and a stepped form. To turn the desired values of matrix tables to zero, in some cases you need to use your imagination and correctly convert their columns or rows. Good luck in mathematics and in working with matrices!
In this topic we will consider the concept of a matrix, as well as types of matrices. Since there are a lot of terms in this topic, I will add a brief summary to make it easier to navigate the material.
Definition of a matrix and its element. Notation.
Matrix is a table of $m$ rows and $n$ columns. The elements of a matrix can be objects of a completely different nature: numbers, variables or, for example, other matrices. For example, the matrix $\left(\begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right)$ contains 3 rows and 2 columns; its elements are integers. The matrix $\left(\begin(array) (cccc) a & a^9+2 & 9 & \sin x \\ -9 & 3t^2-4 & u-t & 8\end(array) \right)$ contains 2 rows and 4 columns.
Different ways to write matrices: show\hide
The matrix can be written not only in round, but also in square or double straight brackets. Below is the same matrix in different notation forms:
$$ \left(\begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right);\;\; \left[ \begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right]; \;\; \left \Vert \begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right \Vert $$
The product $m\times n$ is called matrix size. For example, if a matrix contains 5 rows and 3 columns, then we speak of a matrix of size $5\times 3$. The matrix $\left(\begin(array)(cc) 5 & 3\\0 & -87\\8 & 0\end(array)\right)$ has size $3 \times 2$.
Typically, matrices are denoted by capital letters of the Latin alphabet: $A$, $B$, $C$ and so on. For example, $B=\left(\begin(array) (ccc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right)$. Line numbering goes from top to bottom; columns - from left to right. For example, the first row of matrix $B$ contains elements 5 and 3, and the second column contains elements 3, -87, 0.
Elements of matrices are usually denoted in small letters. For example, the elements of the matrix $A$ are denoted by $a_(ij)$. The double index $ij$ contains information about the position of the element in the matrix. The number $i$ is the row number, and the number $j$ is the column number, at the intersection of which is the element $a_(ij)$. For example, at the intersection of the second row and the fifth column of the matrix $A=\left(\begin(array) (cccccc) 51 & 37 & -9 & 0 & 9 & 97 \\ 1 & 2 & 3 & 41 & 59 & 6 \ \ -17 & -15 & -13 & -11 & -8 & -5 \\ 52 & 31 & -4 & -1 & 17 & 90 \end(array) \right)$ element $a_(25)= $59:
In the same way, at the intersection of the first row and the first column we have the element $a_(11)=51$; at the intersection of the third row and the second column - the element $a_(32)=-15$ and so on. Note that the entry $a_(32)$ reads “a three two”, but not “a thirty two”.
To abbreviate the matrix $A$, the size of which is $m\times n$, the notation $A_(m\times n)$ is used. The following notation is often used:
$$ A_(m\times(n))=(a_(ij)) $$
Here $(a_(ij))$ indicates the designation of the elements of the matrix $A$, i.e. says that the elements of the matrix $A$ are denoted as $a_(ij)$. In expanded form, the matrix $A_(m\times n)=(a_(ij))$ can be written as follows:
$$ A_(m\times n)=\left(\begin(array)(cccc) a_(11) & a_(12) & \ldots & a_(1n) \\ a_(21) & a_(22) & \ldots & a_(2n) \\ \ldots & \ldots & \ldots & \ldots \\ a_(m1) & a_(m2) & \ldots & a_(mn) \end(array) \right) $$
Let's introduce another term - equal matrices.
Two matrices of the same size $A_(m\times n)=(a_(ij))$ and $B_(m\times n)=(b_(ij))$ are called equal, if their corresponding elements are equal, i.e. $a_(ij)=b_(ij)$ for all $i=\overline(1,m)$ and $j=\overline(1,n)$.
Explanation for the entry $i=\overline(1,m)$: show\hide
The notation "$i=\overline(1,m)$" means that the parameter $i$ varies from 1 to m. For example, the notation $i=\overline(1,5)$ indicates that the parameter $i$ takes the values 1, 2, 3, 4, 5.
So, for matrices to be equal, two conditions must be met: coincidence of sizes and equality of the corresponding elements. For example, the matrix $A=\left(\begin(array)(cc) 5 & 3\\0 & -87\\8 & 0\end(array)\right)$ is not equal to the matrix $B=\left(\ begin(array)(cc) 8 & -9\\0 & -87 \end(array)\right)$ because matrix $A$ has size $3\times 2$ and matrix $B$ has size $2\times $2. Also, matrix $A$ is not equal to matrix $C=\left(\begin(array)(cc) 5 & 3\\98 & -87\\8 & 0\end(array)\right)$, since $a_( 21)\neq c_(21)$ (i.e. $0\neq 98$). But for the matrix $F=\left(\begin(array)(cc) 5 & 3\\0 & -87\\8 & 0\end(array)\right)$ we can safely write $A=F$ because both the sizes and the corresponding elements of the matrices $A$ and $F$ coincide.
Example No. 1
Determine the size of the matrix $A=\left(\begin(array) (ccc) -1 & -2 & 1 \\ 5 & 9 & -8 \\ -6 & 8 & 23 \\ 11 & -12 & -5 \ \4 & 0 & -10 \\ \end(array) \right)$. Indicate what the elements $a_(12)$, $a_(33)$, $a_(43)$ are equal to.
This matrix contains 5 rows and 3 columns, so its size is $5\times 3$. You can also use the notation $A_(5\times 3)$ for this matrix.
Element $a_(12)$ is at the intersection of the first row and second column, so $a_(12)=-2$. Element $a_(33)$ is at the intersection of the third row and third column, so $a_(33)=23$. Element $a_(43)$ is at the intersection of the fourth row and third column, so $a_(43)=-5$.
Answer: $a_(12)=-2$, $a_(33)=23$, $a_(43)=-5$.
Types of matrices depending on their size. Main and secondary diagonals. Matrix trace.
Let a certain matrix $A_(m\times n)$ be given. If $m=1$ (the matrix consists of one row), then the given matrix is called matrix-row. If $n=1$ (the matrix consists of one column), then such a matrix is called matrix-column. For example, $\left(\begin(array) (ccccc) -1 & -2 & 0 & -9 & 8 \end(array) \right)$ is a row matrix, and $\left(\begin(array) (c) -1 \\ 5 \\ 6 \end(array) \right)$ is a column matrix.
If the matrix $A_(m\times n)$ satisfies the condition $m\neq n$ (i.e., the number of rows is not equal to the number of columns), then it is often said that $A$ is a rectangular matrix. For example, the matrix $\left(\begin(array) (cccc) -1 & -2 & 0 & 9 \\ 5 & 9 & 5 & 1 \end(array) \right)$ has size $2\times 4$, those. contains 2 rows and 4 columns. Since the number of rows is not equal to the number of columns, this matrix is rectangular.
If the matrix $A_(m\times n)$ satisfies the condition $m=n$ (i.e., the number of rows is equal to the number of columns), then $A$ is said to be a square matrix of order $n$. For example, $\left(\begin(array) (cc) -1 & -2 \\ 5 & 9 \end(array) \right)$ is a second-order square matrix; $\left(\begin(array) (ccc) -1 & -2 & 9 \\ 5 & 9 & 8 \\ 1 & 0 & 4 \end(array) \right)$ is a third-order square matrix. In general, the square matrix $A_(n\times n)$ can be written as follows:
$$ A_(n\times n)=\left(\begin(array)(cccc) a_(11) & a_(12) & \ldots & a_(1n) \\ a_(21) & a_(22) & \ldots & a_(2n) \\ \ldots & \ldots & \ldots & \ldots \\ a_(n1) & a_(n2) & \ldots & a_(nn) \end(array) \right) $$
The elements $a_(11)$, $a_(22)$, $\ldots$, $a_(nn)$ are said to be on main diagonal matrices $A_(n\times n)$. These elements are called main diagonal elements(or just diagonal elements). The elements $a_(1n)$, $a_(2 \; n-1)$, $\ldots$, $a_(n1)$ are on side (minor) diagonal; they are called side diagonal elements. For example, for the matrix $C=\left(\begin(array)(cccc)2&-2&9&1\\5&9&8& 0\\1& 0 & 4 & -7 \\ -4 & -9 & 5 & 6\end(array) \right)$ we have:
The elements $c_(11)=2$, $c_(22)=9$, $c_(33)=4$, $c_(44)=6$ are the main diagonal elements; elements $c_(14)=1$, $c_(23)=8$, $c_(32)=0$, $c_(41)=-4$ are side diagonal elements.
The sum of the main diagonal elements is called followed by the matrix and is denoted by $\Tr A$ (or $\Sp A$):
$$ \Tr A=a_(11)+a_(22)+\ldots+a_(nn) $$
For example, for the matrix $C=\left(\begin(array) (cccc) 2 & -2 & 9 & 1\\5 & 9 & 8 & 0\\1 & 0 & 4 & -7\\-4 & -9 & 5 & 6 \end(array)\right)$ we have:
$$ \Tr C=2+9+4+6=21. $$
The concept of diagonal elements is also used for non-square matrices. For example, for the matrix $B=\left(\begin(array) (ccccc) 2 & -2 & 9 & 1 & 7 \\ 5 & -9 & 8 & 0 & -6 \\ 1 & 0 & 4 & - 7 & -6 \end(array) \right)$ the main diagonal elements will be $b_(11)=2$, $b_(22)=-9$, $b_(33)=4$.
Types of matrices depending on the values of their elements.
If all elements of the matrix $A_(m\times n)$ are equal to zero, then such a matrix is called null and is usually denoted by the letter $O$. For example, $\left(\begin(array) (cc) 0 & 0 \\ 0 & 0 \\ 0 & 0 \end(array) \right)$, $\left(\begin(array) (ccc) 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end(array) \right)$ - zero matrices.
Let's consider some non-zero row of the matrix $A$, i.e. a string that contains at least one element other than zero. Leading element of a non-zero string we call its first (counting from left to right) non-zero element. For example, consider the following matrix:
$$W=\left(\begin(array)(cccc) 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 12\\ 0 & -9 & 5 & 9 \end(array)\right)$ $
In the second line the leading element will be the fourth element, i.e. $w_(24)=12$, and in the third line the leading element will be the second element, i.e. $w_(32)=-9$.
The matrix $A_(m\times n)=\left(a_(ij)\right)$ is called stepped, if it satisfies two conditions:
- Null rows, if present, are located below all non-null rows.
- The numbers of the leading elements of non-zero rows form a strictly increasing sequence, i.e. if $a_(1k_1)$, $a_(2k_2)$, ..., $a_(rk_r)$ are the leading elements of non-zero rows of the matrix $A$, then $k_1\lt(k_2)\lt\ldots\lt( k_r)$.
Examples of step matrices:
$$ \left(\begin(array)(cccccc) 0 & 0 & 2 & 0 & -4 & 1\\ 0 & 0 & 0 & 0 & -9 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 \end(array)\right);\; \left(\begin(array)(cccc) 5 & -2 & 2 & -8\\ 0 & 4 & 0 & 0\\ 0 & 0 & 0 & -10 \end(array)\right). $$
For comparison: matrix $Q=\left(\begin(array)(ccccc) 2 & -2 & 0 & 1 & 9\\0 & 0 & 0 & 7 & 9\\0 & -5 & 0 & 10 & 6\end(array)\right)$ is not a step matrix, since the second condition in the definition of a step matrix is violated. The leading elements in the second and third rows $q_(24)=7$ and $q_(32)=10$ have numbers $k_2=4$ and $k_3=2$. For a step matrix, the condition $k_2\lt(k_3)$ must be satisfied, which is violated in this case. Let me note that if we swap the second and third rows, we get a stepwise matrix: $\left(\begin(array)(ccccc) 2 & -2 & 0 & 1 & 9\\0 & -5 & 0 & 10 & 6 \\0 & 0 & 0 & 7 & 9\end(array)\right)$.
A step matrix is called trapezoidal or trapezoidal, if the leading elements $a_(1k_1)$, $a_(2k_2)$, ..., $a_(rk_r)$ satisfy the conditions $k_1=1$, $k_2=2$,..., $k_r= r$, i.e. the leading ones are the diagonal elements. In general, a trapezoidal matrix can be written as follows:
$$ A_(m\times(n)) =\left(\begin(array) (cccccc) a_(11) & a_(12) & \ldots & a_(1r) & \ldots & a_(1n)\\ 0 & a_(22) & \ldots & a_(2r) & \ldots & a_(2n)\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & \ldots & a_(rr) & \ldots & a_(rn)\\ 0 & 0 & \ldots & 0 & \ldots & 0\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & \ldots & 0 & \ldots & 0 \end(array)\right) $$
Examples of trapezoidal matrices:
$$ \left(\begin(array)(cccccc) 4 & 0 & 2 & 0 & -4 & 1\\ 0 & -2 & 0 & 0 & -9 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 \end(array)\right);\; \left(\begin(array)(cccc) 5 & -2 & 2 & -8\\ 0 & 4 & 0 & 0\\ 0 & 0 & -3 & -10 \end(array)\right). $$
Let's give a few more definitions for square matrices. If all elements of a square matrix located under the main diagonal are equal to zero, then such a matrix is called upper triangular matrix. For example, $\left(\begin(array) (cccc) 2 & -2 & 9 & 1 \\ 0 & 9 & 8 & 0 \\ 0 & 0 & 4 & -7 \\ 0 & 0 & 0 & 6 \end(array) \right)$ is an upper triangular matrix. Note that the definition of an upper triangular matrix does not say anything about the values of the elements located above the main diagonal or on the main diagonal. They can be zero or not - it doesn't matter. For example, $\left(\begin(array) (ccc) 0 & 0 & 9 \\ 0 & 0 & 0\\ 0 & 0 & 0 \end(array) \right)$ is also an upper triangular matrix.
If all elements of a square matrix located above the main diagonal are equal to zero, then such a matrix is called lower triangular matrix. For example, $\left(\begin(array) (cccc) 3 & 0 & 0 & 0 \\ -5 & 1 & 0 & 0 \\ 8 & 2 & 1 & 0 \\ 5 & 4 & 0 & 6 \ end(array) \right)$ - lower triangular matrix. Note that the definition of a lower triangular matrix does not say anything about the values of the elements located under or on the main diagonal. They may be zero or not - it doesn't matter. For example, $\left(\begin(array) (ccc) -5 & 0 & 0 \\ 0 & 0 & 0\\ 0 & 0 & 9 \end(array) \right)$ and $\left(\begin (array) (ccc) 0 & 0 & 0 \\ 0 & 0 & 0\\ 0 & 0 & 0 \end(array) \right)$ are also lower triangular matrices.
The square matrix is called diagonal, if all elements of this matrix that do not lie on the main diagonal are equal to zero. Example: $\left(\begin(array) (cccc) 3 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 \ end(array)\right)$. The elements on the main diagonal can be anything (equal to zero or not) - it doesn't matter.
The diagonal matrix is called single, if all elements of this matrix located on the main diagonal are equal to 1. For example, $\left(\begin(array) (cccc) 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end(array)\right)$ - fourth-order identity matrix; $\left(\begin(array) (cc) 1 & 0 \\ 0 & 1 \end(array)\right)$ is the second-order identity matrix.