Matrix size m ? n called rectangular table numbers containing m rows and n columns. The numbers that make up the matrix are called elements matrices.
The matrices are denoted in capital letters Latin alphabet (A,B,C...), and to designate matrix elements we use lowercase letters with double indexing:
Where i- line number, j- column number.
For example, matrix
Or in shorthand, A=(); i=1,2…, m; j=1,2, …, n.
Other matrix notations are used, for example: , ? ?.
Two matrices A And IN same size are called equal, if they coincide element by element, i.e. = , where i= 1, 2, 3, …, m, A j= 1, 2, 3, …, n.
Let's consider the main types of matrices:
1. Let m = n, then matrix A is a square matrix that has order n:
The elements form the main diagonal, the elements form the secondary diagonal.
The square matrix is called diagonal, if all its elements, except perhaps the elements of the main diagonal, are equal to zero:
A diagonal, and therefore square, matrix is called single, if all elements of the main diagonal are equal to 1:
Note that the identity matrix is the matrix analogue of the unit in the set real numbers, and also emphasize that the identity matrix is defined only for square matrices.
Here are examples of identity matrices:
Square matrices
are called upper and lower triangular, respectively.
- 2. Let m= 1, then the matrix A- row matrix, which looks like:
- 3. Let n=1, then the matrix A- column matrix, which looks like:
4. A zero matrix is a matrix of order mn, all elements of which are equal to 0:
Note that the null matrix can be a square matrix, a row matrix, or a column matrix. The zero matrix is the matrix analogue of zero in the set of real numbers.
5. A matrix is called transposed to a matrix and is denoted if its columns are the rows of the matrix corresponding in number.
Example. Let
Note that if the matrix A has order mn, then the transposed matrix has the order nm.
6. Matrix A is called symmetric if A =, and skew-symmetric if A =.
Example. Examine for matrix symmetry A And IN.
hence the matrix A- symmetrical, because A =.
hence the matrix IN- skew-symmetrical, since B = -.
Note that symmetric and skew-symmetric matrices are always square. Any elements can be on the main diagonal of a symmetric matrix, and identical elements must be placed symmetrically relative to the main diagonal, that is, zeros always appear on the main diagonal of a skew-symmetric matrix, and symmetrically relative to the main diagonal
matrix square laplace cancellation
1st year, higher mathematics, studying matrices and basic actions on them. Here we systematize the basic operations that can be performed with matrices. Where to start getting acquainted with matrices? Of course, from the simplest things - definitions, basic concepts and simple operations. We assure you that the matrices will be understood by everyone who devotes at least a little time to them!
Matrix Definition
Matrix is a rectangular table of elements. Well, what if in simple language– table of numbers.
Typically matrices are denoted in capitals in Latin letters. For example, matrix A , matrix B and so on. Matrices can be different sizes: rectangular, square, there are also row matrices and column matrices called vectors. The size of the matrix is determined by the number of rows and columns. For example, let's write rectangular matrix size m on n , Where m – number of lines, and n – number of columns.
Items for which i=j (a11, a22, .. ) form the main diagonal of the matrix and are called diagonal.
What can you do with matrices? Add/Subtract, multiply by a number, multiply among themselves, transpose. Now about all these basic operations on matrices in order.
Matrix addition and subtraction operations
Let us immediately warn you that you can only add matrices same size. The result will be a matrix of the same size. Adding (or subtracting) matrices is simple - you just need to add up their corresponding elements . Let's give an example. Let's perform the addition of two matrices A and B of size two by two.
Subtraction is performed by analogy, only with the opposite sign.
On arbitrary number You can multiply any matrix. To do this you need to multiply each of its elements by this number. For example, let's multiply the matrix A from the first example by the number 5:
Matrix multiplication operation
Not all matrices can be multiplied together. For example, we have two matrices - A and B. They can be multiplied by each other only if the number of columns of matrix A is equal to the number of rows of matrix B. In this case each element of the resulting matrix located in the i-th row and jth column, will equal to the sum products of the corresponding elements in i-th line the first factor and the j-th column of the second. To understand this algorithm, let's write down how two square matrices are multiplied:
And an example with real numbers. Let's multiply the matrices:
Matrix transpose operation
Matrix transposition is an operation where the corresponding rows and columns are swapped. For example, let's transpose the matrix A from the first example:
Matrix determinant
Determinant, or determinant, is one of the basic concepts linear algebra. Once upon a time, people came up with linear equations, and after them they had to come up with a determinant. In the end, it’s up to you to deal with all this, so, the last push!
The determinant is numerical characteristic square matrix, which is needed to solve many problems.
To calculate the determinant of the simplest square matrix, you need to calculate the difference between the products of the elements of the main and secondary diagonals.
The determinant of a matrix of first order, that is, consisting of one element, is equal to this element.
What if the matrix is three by three? This is more difficult, but you can manage it.
For such a matrix, the value of the determinant is equal to the sum of the products of the elements of the main diagonal and the products of the elements lying on the triangles with a face parallel to the main diagonal, from which the product of the elements of the secondary diagonal and the product of the elements lying on the triangles with the face of the parallel secondary diagonal are subtracted.
Fortunately, computing determinants of matrices large sizes in practice it is rarely necessary.
Here we looked at basic operations on matrices. Of course, in real life you may never encounter even a hint of matrix system equations or vice versa - face much more complex cases when you really have to rack your brains. It is for such cases that professional student services exist. Ask for help, get quality and detailed solution, enjoy your academic success and free time.
A matrix is a special object in mathematics. Shown in a rectangular or square table, made up of a certain number 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 linear equations and convenient 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 in many other ways. The basic skill when working with matrices is reduction to standard view. 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 include the existence of a second-order matrix (2x2 matrix), fourth order(4x4), tenth (10x10), seventeenth (17x17) and so on.
Column vector
This is one of the simplest types of matrices, containing only one column, which includes three numerical values. She represents a series free members(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 components of the main diagonal (highlighted green). The main diagonal begins with the element located in the upper right corner and ends with the number in the third column of the third row. 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 same values.
A subtype of diagonal matrix. All of her numeric 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; Reducing to it is often necessary for work. Number of rows and columns in canonical matrix different, it does not necessarily belong to square type. It is somewhat similar to the identity matrix, but in its case not all components of the main diagonal take on the 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 view is necessary to find the rank of a matrix, as well as for elementary operations on them (along with 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. The prerequisite is the following: if in step matrix If there is a zero line, then the remaining lines below it also do not contain numeric values.
So we looked at most important types 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.
Task 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 convert it to triangular shape we need to turn two components of the first column and one component of the second to zero.
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.
All that's left is last value- 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 converting it from the element located in the lower left corner - from the number 4. We need to reverse given 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 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: just add to last line the third and we 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 calmly begin 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.
Task 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. He 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 make it zero required values matrix tables, in 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!
Definition by Matrix– called a table of numbers containing a certain number of rows and columns
The elements of the matrix are numbers of the form a ij, where i is the row number j is the column number
Example 1 i = 2 j = 3
Designation: A=
Types of matrices:
1. If the number of rows is not equal to the number of columns, then the matrix is called rectangular:
2. If the number of rows is equal to the number of columns, then the matrix is called square:
The number of rows or columns of a square matrix is called its in order. In the example n = 2
Consider a square matrix of order n:
The diagonal containing the elements a 11, a 22......., a nn is called main , and the diagonal containing the elements a 12, a 2 n -1, …….a n 1 – auxiliary.
A matrix in which only the elements on the main diagonal are nonzero is called diagonal:
Example 4 
n=3
3. If a diagonal matrix has elements equal to 1, then the matrix is called single and is designated by the letter E:
Example 6 
n=3
4. A matrix, all elements of which are equal to zero, is called null matrix and is denoted by the letter O
Example 7 
5. Triangular A matrix of nth order is a square matrix, all of whose elements located below the main diagonal are equal to zero:
Example 8 
n=3
Actions on matrices:
The sum of a matrix A and B is a matrix C whose elements are equal to the sum of the corresponding elements of matrices A and B.
Only matrices that have same number rows and columns.
Product of matrix A and number k such a matrix kA is called, each element of which is equal to ka ij
Example10 
Multiplying a matrix by a number is reduced to multiplying all the elements of the matrix by that number.
Product of matrices To multiply a matrix by a matrix, you need to select the first row of the first matrix and multiply by the corresponding elements of the first column of the second matrix, and add the result. Place this result in the result matrix in the 1st row and 10th column. We perform the same actions with all other elements: 1st line to the second column, to the 3rd, etc., then with the following lines.
Example 11 
Multiplying matrix A by matrix B is possible only if the number of columns of the first matrix is equal to the number of columns of the second matrix.
- the work exists;
- the work does not exist
Examples 12 
there is nothing to multiply the last line in matrix II with, i.e. the work does not exist
Matrix Transpose The operation of replacing row elements with column elements is called:
Example13 
By raising to a power is called sequential multiplication of a matrix by itself.
Let there be a square matrix of nth order
Matrix A -1 is called inverse matrix in relation to matrix A, if A*A -1 = E, where E is the identity matrix of the nth order.
Identity matrix- such a square matrix in which all elements are along the main diagonal passing from the left top corner to the lower right corner are ones, and the rest are zeros, for example:
Inverse matrix may exist only for square matrices those. for those matrices in which the number of rows and columns coincide.
Theorem for the existence condition of an inverse matrix
In order for a matrix to have an inverse matrix, it is necessary and sufficient that it be non-singular.
The matrix A = (A1, A2,...A n) is called non-degenerate, if the column vectors are linearly independent. The number of linearly independent column vectors of a matrix is called the rank of the matrix. Therefore, we can say that in order for an inverse matrix to exist, it is necessary and sufficient that the rank of the matrix is equal to its dimension, i.e. r = n.
Algorithm for finding the inverse matrix
- Write matrix A into the table for solving systems of equations using the Gaussian method and assign matrix E to it on the right (in place of the right-hand sides of the equations).
- Using Jordan transformations, reduce matrix A to a matrix consisting of unit columns; in this case, it is necessary to simultaneously transform the matrix E.
- If necessary, rearrange the rows (equations) of the last table so that under the matrix A of the original table you get the identity matrix E.
- Write down the inverse matrix A -1, which is in last table under matrix E of the original table.
For matrix A, find the inverse matrix A -1
Solution: We write matrix A and assign the identity matrix E to the right. Using Jordan transformations, we reduce matrix A to the identity matrix E. The calculations are given in Table 31.1.
Let's check the correctness of the calculations by multiplying the original matrix A and inverse matrix A -1.
As a result of matrix multiplication, the identity matrix was obtained. Therefore, the calculations were made correctly.
Answer: 
Solving matrix equations
Matrix equations can look like:
AX = B, HA = B, AXB = C,
where A, B, C are the specified matrices, X is the desired matrix.
Matrix equations are solved by multiplying the equation by inverse matrices.
For example, to find the matrix from the equation, you need to multiply this equation by on the left.
Therefore, to find a solution to the equation, you need to find the inverse matrix and multiply it by the matrix on the right side of the equation.
Other equations are solved similarly.
Solve the equation AX = B if 
Solution: Since the inverse matrix is equal to (see example 1)
Matrix method in economic analysis
Along with others, they are also used matrix methods . These methods are based on linear and vector-matrix algebra. Such methods are used for the purposes of analyzing complex and multidimensional economic phenomena. Most often, these methods are used when it is necessary to make a comparative assessment of the functioning of organizations and their structural divisions.
In the process of applying matrix analysis methods, several stages can be distinguished.
At the first stage the system is being formed economic indicators and on its basis, a source data matrix is compiled, which is a table in which system numbers are shown in its individual rows (i = 1,2,....,n), and in vertical columns - numbers of indicators (j = 1,2,....,m).
At the second stage For each vertical column, the largest of the available indicator values is identified, which is taken as one.
After this, all amounts reflected in this column are divided by highest value and a matrix of standardized coefficients is formed.
At the third stage all components of the matrix are squared. If they have different significance, then each matrix indicator is assigned a certain weight coefficient k. The value of the latter is determined by expert opinion.
On the last one, fourth stage found quantities ratings Rj are grouped in order of their increase or decrease.
The matrix methods outlined should be used, for example, when comparative analysis various investment projects, as well as when assessing other economic indicators of organizations.
