A mathematical matrix is a table of ordered elements. The dimensions of this table are determined by the number of rows and columns in it. As for the solution of matrices, they call a huge number of operations that are performed on these same matrices. Mathematicians distinguish several types of matrices. For some of them, the general rules for the decision apply, while for others they do not. For example, if the matrices have the same dimension, then they can be added, and if they are consistent with each other, then they can be multiplied. It is necessary to find a determinant to solve any matrix. In addition, matrices are subject to transposition and finding minors in them. So let's look at how to solve matrices.
Order of solving matrices
First, we write down the given matrices. We count how many rows and columns they have. If the number of rows and columns is the same, then such a matrix is called square. If each element of the matrix is zero, then this matrix is zero. The next thing we do is find the main diagonal of the matrix. The elements of such a matrix are from the lower right corner to the upper left. The second diagonal in the matrix is a side diagonal. Now we need to transpose the matrix. To do this, it is necessary to replace the row elements in each of the two matrices with the corresponding column elements. For example, the element under a21 will be the element a12, or vice versa. Thus, after this procedure, a completely different matrix should appear.
If the matrices have exactly the same dimension, then they can be easily added. To do this, we take the first element of the first matrix a11 and add it to the similar element of the second matrix b11. What happens as a result, we write to the same position, only in a new matrix. Now we add all the other elements of the matrix in the same way until we get a new completely different matrix. Let's see a few more ways to solve matrices.
Options for actions with matrices
We can also determine if the matrices are consistent. To do this, we need to compare the number of rows in the first matrix with the number of columns in the second matrix. If they are equal, you can multiply them. To do this, we pairwise multiply an element in a row of one matrix by a similar element in a column of another matrix. Only after that it will be possible to calculate the sum of the resulting products. Based on this, the initial element of the matrix that should be obtained as a result will be equal to g11 = a11 * b11 + a12 * b21 + a13 * b31 + ... + a1m * bn1. After the addition and multiplication of all products is completed, you can fill in the final matrix.
It is also possible, when solving matrices, to find their determinant and determinant for each. If the matrix is square and has a dimension of 2 by 2, then the determinant can be found as the difference of all products of the elements of the main and secondary diagonals. If the matrix is already three-dimensional, then the determinant can be found by applying the following formula. D \u003d a11 * a22 * a33 + a13 * a21 * a32 + a12 * a23 * a31 - a21 * a12 * a33 - a13 * a22 * a31 - a11 * a32 * a23.
To find the minor of a given element, you need to cross out the column and row where this element is located. Then find the determinant of this matrix. He will be the corresponding minor. A similar decision matrix method was developed several decades ago in order to increase the reliability of the result by dividing the problem into subproblems. Thus, solving matrices is not that difficult if you know the basic mathematical operations.
Matrices. Actions on matrices. Properties of operations on matrices. Types of matrices.
Matrices (and accordingly the mathematical section - matrix algebra) are important in applied mathematics, as they allow writing in a fairly simple form a significant part of the mathematical models of objects and processes. The term "matrix" appeared in 1850. Matrices were first mentioned in ancient China, later by Arab mathematicians.
Matrix A=Amn order m*n is called rectangular table of numbers containing m - rows and n - columns.
Matrix elements aij , for which i=j are called diagonal and form main diagonal.
For a square matrix (m=n), the main diagonal is formed by the elements a 11 , a 22 ,..., a nn .
Matrix equality.
A=B, if the matrix orders A And B are the same and a ij =b ij (i=1,2,...,m; j=1,2,...,n)
Actions on matrices.
1. Matrix addition - element-wise operation
2. Matrix subtraction - element by element operation
3. The product of a matrix by a number is an element-by-element operation
4. Multiplication A*B matrices according to the rule row per column(the number of columns of matrix A must be equal to the number of rows of matrix B)
A mk *B kn =C mn and each element with ij matrices Cmn is equal to the sum of the products of the elements of the i-th row of matrix A by the corresponding elements of the j-th column of matrix B, i.e.
Let's show the operation of matrix multiplication using an example
5. Exponentiation
m>1 is a positive integer. A is a square matrix (m=n) i.e. relevant only for square matrices
6. Transposition of matrix A. The transposed matrix is denoted A T or A "
Rows and columns are swapped
Example
Properties of operations on matrices
(A+B)+C=A+(B+C)
λ(A+B)=λA+λB
A(B+C)=AB+AC
(A+B)C=AC+BC
λ(AB)=(λA)B=A(λB)
A(BC)=(AB)C
(λA)"=λ(A)"
(A+B)"=A"+B"
(AB)"=B"A"
Types of matrices
1. Rectangular: m And n- arbitrary positive integers
2. Square: m=n
3. Matrix row: m=1. For example, (1 3 5 7) - in many practical problems such a matrix is called a vector
4. Matrix column: n=1. For example
5. Diagonal Matrix: m=n And a ij =0, if i≠j. For example
6. Identity matrix: m=n And
7. Zero matrix: a ij =0, i=1,2,...,m
j=1,2,...,n
8. Triangular matrix: all elements below the main diagonal are 0.
9. Symmetric matrix: m=n And aij=aji(i.e., there are equal elements on places that are symmetrical with respect to the main diagonal), and therefore A"=A
For example,
10. Skew matrix: m=n And a ij =-a ji(i.e. opposite elements stand on places that are symmetrical with respect to the main diagonal). Therefore, there are zeros on the main diagonal (because at i=j we have a ii =-a ii)
It's clear, A"=-A
11. Hermitian matrix: m=n And a ii =-ã ii (ã ji- complex - conjugate to a ji, i.e. if A=3+2i, then the complex conjugate Ã=3-2i)
This is a concept that generalizes all possible operations performed with matrices. Mathematical matrix - a table of elements. About a table where m lines and n columns, they say that this matrix has the dimension m on the n.
General view of the matrix:
For matrix solutions you need to understand what a matrix is and know its main parameters. The main elements of the matrix:
- Main Diagonal Consisting of Elements a 11, a 22 ..... a mn.
- Side diagonal consisting of elements а 1n ,а 2n-1 …..а m1.
The main types of matrices:
- Square - such a matrix, where the number of rows = the number of columns ( m=n).
- Zero - where all elements of the matrix = 0.
- Transposed Matrix - Matrix IN, which was obtained from the original matrix A by replacing rows with columns.
- Single - all elements of the main diagonal = 1, all others = 0.
- An inverse matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix.
The matrix can be symmetrical with respect to the main and secondary diagonals. That is, if a 12 = a 21, a 13 \u003d a 31, .... a 23 \u003d a 32 .... a m-1n =a mn-1, then the matrix is symmetric with respect to the main diagonal. Only square matrices can be symmetric.
Methods for solving matrices.
Almost all matrix solution methods are to find its determinant n th order and most of them are quite cumbersome. To find the determinant of the 2nd and 3rd order, there are other, more rational ways.
Finding determinants of the 2nd order.
To calculate the matrix determinant BUT 2nd order, it is necessary to subtract the product of the elements of the secondary diagonal from the product of the elements of the main diagonal:
Methods for finding determinants of the 3rd order.
Below are the rules for finding the 3rd order determinant.
Simplified the triangle rule as one of matrix solution methods, can be represented as follows:
In other words, the product of elements in the first determinant that are connected by lines is taken with a "+" sign; also, for the 2nd determinant - the corresponding products are taken with the "-" sign, that is, according to the following scheme:
At solving matrices by the Sarrus rule, to the right of the determinant, the first 2 columns are added and the products of the corresponding elements on the main diagonal and on the diagonals that are parallel to it are taken with a "+" sign; and the products of the corresponding elements of the secondary diagonal and the diagonals that are parallel to it, with the sign "-":
Row or column expansion of determinant when solving matrices.
The determinant is equal to the sum of the products of the elements of the row of the determinant and their algebraic complements. Usually choose the row/column in which/th there are zeros. The row or column on which the decomposition is carried out will be indicated by an arrow.
Reducing the determinant to a triangular form when solving matrices.
At solving matrices By reducing the determinant to a triangular form, they work like this: using the simplest transformations on rows or columns, the determinant becomes triangular and then its value, in accordance with the properties of the determinant, will be equal to the product of the elements that stand on the main diagonal.
Laplace's theorem for solving matrices.
When solving matrices using Laplace's theorem, it is necessary to know the theorem itself directly. Laplace's theorem: Let Δ is a determinant n-th order. We select any k rows (or columns), provided k≤ n - 1. In this case, the sum of the products of all minors k th order contained in the selected k rows (columns), their algebraic additions will be equal to the determinant.
Inverse matrix solution.
Sequence of actions for inverse matrix solutions:
- Find out if the given matrix is square. In the case of a negative answer, it becomes clear that there cannot be an inverse matrix for it.
- We calculate algebraic additions.
- We compose the allied (mutual, attached) matrix C.
- We compose an inverse matrix from algebraic additions: all elements of the adjoint matrix C divide by the determinant of the initial matrix. The resulting matrix will be the desired inverse matrix with respect to the given one.
- We check the work done: we multiply the matrix of the initial and the resulting matrices, the result should be the identity matrix.
Solution of matrix systems.
For solutions of matrix systems most commonly used is the Gauss method.
The Gauss method is a standard way to solve systems of linear algebraic equations (SLAE) and it consists in the fact that variables are sequentially eliminated, i.e., with the help of elementary changes, the system of equations is brought to an equivalent system of a triangular form and from it, sequentially, starting from the last (by number), find each element of the system.
Gauss method is the most versatile and best tool for finding matrix solutions. If the system has an infinite number of solutions or the system is incompatible, then it cannot be solved using Cramer's rule and the matrix method.
The Gauss method also implies direct (reduction of the extended matrix to a stepped form, i.e. getting zeros under the main diagonal) and reverse (getting zeros over the main diagonal of the extended matrix) moves. The forward move is the Gauss method, the reverse is the Gauss-Jordan method. The Gauss-Jordan method differs from the Gauss method only in the sequence of elimination of variables.
Matrices in mathematics are one of the most important objects of applied importance. Often an excursion into the theory of matrices begins with the words: "A matrix is a rectangular table ...". We will start this excursion from a slightly different angle.
Phone books of any size and with any number of subscriber data are nothing but matrices. These matrices look like this:
It is clear that we all use such matrices almost every day. These matrices come in various numbers of rows (distinguished as a directory issued by the telephone company, which may contain thousands, hundreds of thousands, and even millions of lines, and a new notebook you just started, which has less than ten lines) and columns (a directory of officials of some some organization in which there may be such columns as the position and office number and the same your notebook, where there may be no data other than the name, and, thus, it has only two columns - name and phone number).
All matrices can be added and multiplied, and other operations can be performed on them, but there is no need to add and multiply telephone directories, there is no benefit from this, and besides, you can move your mind.
But very many matrices can and should be added and multiplied and various urgent tasks can be solved in this way. Below are examples of such matrices.
Matrices in which the columns are the output of units of a particular type of product, and the rows are the years in which the output of this product is recorded:
You can add matrices of this kind, which take into account the output of similar products by various enterprises, in order to obtain summary data for the industry.
Or matrices, consisting, for example, of one column, in which the rows are the average cost of a particular type of product:
Matrices of the last two types can be multiplied, and the result is a row matrix containing the cost of all types of products by years.
Matrices, basic definitions
Rectangular table consisting of numbers arranged in m lines and n columns is called mn-matrix (or simply matrix ) and written like this:
(1)
In matrix (1) the numbers are called its elements (as in the determinant, the first index means the number of the row, the second - the column, at the intersection of which there is an element; i = 1, 2, ..., m; j = 1, 2, n).
The matrix is called rectangular , if .
If m = n, then the matrix is called square , and the number n is its in order .
The determinant of the square matrix A is called the determinant whose elements are the elements of the matrix A. It is denoted by the symbol | A|.
The square matrix is called non-special (or non-degenerate , non-singular ) if its determinant is not equal to zero, and special (or degenerate , singular ) if its determinant is zero.
The matrices are called equal if they have the same number of rows and columns and all matching elements are the same.
The matrix is called null if all its elements are equal to zero. The zero matrix will be denoted by the symbol 0 or .
For example,
row matrix (or lowercase ) is called 1 n-matrix, and column matrix (or columnar ) – m 1-matrix.
The matrix A" , which is obtained from the matrix A swapping rows and columns in it is called transposed with respect to the matrix A. Thus, for matrix (1), the transposed matrix is
Transition to matrix operation A" , transposed with respect to the matrix A, is called the transposition of the matrix A. For mn-matrix transposed is nm-the matrix.
The matrix transposed with respect to the matrix is A, i.e
(A")" = A .
Example 1 Find Matrix A" , transposed with respect to the matrix
and find out if the determinants of the original and transposed matrices are equal.
main diagonal A square matrix is an imaginary line connecting its elements, for which both indices are the same. These elements are called diagonal .
A square matrix in which all elements outside the main diagonal are equal to zero is called diagonal . Not all diagonal elements of a diagonal matrix are necessarily nonzero. Some of them may be equal to zero.
A square matrix in which the elements on the main diagonal are equal to the same non-zero number, and all others are equal to zero, is called scalar matrix .
identity matrix is called a diagonal matrix in which all diagonal elements are equal to one. For example, the identity matrix of the third order is the matrix
Example 2 Matrix data:
Solution. Let us calculate the determinants of these matrices. Using the rule of triangles, we find
Matrix determinant B calculate by the formula 
We easily get that 
Therefore, the matrices A and are non-singular (non-degenerate, non-singular), and the matrix B- special (degenerate, singular).
The determinant of an identity matrix of any order is obviously equal to one.
Solve the matrix problem yourself, and then see the solution
Example 3 Matrix data
,
,
Determine which of them are non-singular (non-degenerate, non-singular).
Application of matrices in mathematical and economic modeling
In the form of matrices, structured data about a particular object is simply and conveniently written. Matrix models are created not only to store this structured data, but also to solve various problems with this data using linear algebra.
Thus, the well-known matrix model of the economy is the input-output model introduced by the American economist of Russian origin Wassily Leontiev. This model is based on the assumption that the entire manufacturing sector of the economy is divided into n clean industries. Each of the industries produces only one type of product and different industries produce different products. Because of this division of labor between industries, there are inter-industry relations, the meaning of which is that part of the production of each industry is transferred to other industries as a production resource.
Production volume i-th industry (measured by a specific unit of measure) that was produced during the reporting period, denoted by and called the total output i th industry. Issues are conveniently placed in n-component row of the matrix.
Number of product units i-th industry to be spent j-th industry for the production of a unit of its output, is denoted and called the coefficient of direct costs.
Instruction
The number of columns and rows is set dimension matrices. For instance, dimension yu 5x6 has 5 rows and 6 columns. In general, dimension matrices is written as m×n, where the number m indicates the number of rows, n - columns.
If the array has dimension m×n, it can be multiplied by an n×l array. Number of columns first matrices must equal the number of rows of the second, otherwise the multiplication operation will not be defined.
Dimension matrices indicates the number of equations in the system and the number of variables. The number of rows matches the number of equations, and each column has its own variable. The solution of the system of linear equations is "recorded" in operations on matrices. Thanks to the matrix notation system, high-order systems are possible.
If the number of rows is equal to the number of columns, the matrix is square. It contains the main and secondary diagonals. The main one goes from the upper left corner to the lower right, the secondary one goes from the upper right to the lower left.
Arrays dimension and m×1 or 1×n are vectors. You can also represent any row and any column of an arbitrary table as a vector. For such matrices, all operations on vectors are defined.
In programming, a rectangular table is given two indexes, one of which runs through the entire row, the other - the length of the column. In this case, the cycle for one index is placed inside the cycle for another, due to which the sequential passage of the entire dimension matrices.
matrices is an efficient way of representing numerical information. The solution of any system of linear equations can be written as a matrix (a rectangle made up of numbers). The ability to multiply matrices is one of the most important skills taught in the Linear Algebra course in higher education.
You will need
- Calculator
Instruction
To check this condition, the easiest way is to use the following algorithm - write down the dimension of the first matrix as (a*b). Further dimension of the second - (c*d). If b=c - matrices are commensurate, they can be multiplied.
Then do the multiplication itself. Remember - when you multiply two matrices, you get a matrix. That is, the problem of multiplication is reduced to the problem of finding a new one, with dimension (a * d). In SI, the problem of matrix multiplication looks like this:
void matrixmult(int m1[n], int m1_row, int m1_col, int m2[n], int m2_row, int m2_col, int m3[n], int m3_row, int m3_col)
( for (int i = 0; i< m3_row; i++)
for (int j = 0; j< m3_col; j++)
m3[i][j]=0;
for (int k = 0; k< m2_col; k++)
for (int i = 0; i< m1_row; i++)
for (int j = 0; j< m1_col; j++)
m3[i][k] += m1[i][j] * m2[j][k];
}
Simply put, the new matrix is the sum of the products of the elements of the row of the first matrix by the elements of the column of the second matrix. If you are an element of the third matrix with number (1;2), then you should simply multiply the first row of the first matrix by the second column of the second. To do this, consider the initial amount equal to zero. Then multiply the first element of the first row by the first element of the second column, add the value to the sum. You do this: you multiply the i-th element of the first row by the i-th element of the second column and add the results to the sum until the line ends. The final sum will be the desired element.
After you have found all the elements of the third matrix, write it down. You have found work matrices.
Sources:
- The main mathematical portal of Russia in 2019
- how to find product of matrices in 2019
A mathematical matrix is an ordered table of elements. Dimension matrices is determined by the number of its rows m and columns n. Matrix solving is understood as a set of generalizing operations performed on matrices. There are several types of matrices, some of them are not applicable to a number of operations. There is an addition operation for matrices with the same dimension. The product of two matrices is found only if they are compatible. For any matrices determinant is determined. Also, the matrix can be transposed and the minor of its elements can be determined.
Instruction
Write down the assignments. Determine their dimensions. To do this, count the number of columns n and rows m. If for one matrices m = n, the matrix is assumed to be square. If all elements matrices equal to zero, the matrix is zero. Determine the main diagonal of the matrices. Its elements are located from the upper left corner matrices to the lower right. Second, reverse diagonal matrices is a side.
Perform a matrix transposition. To do this, replace in each row elements with column elements relative to the main diagonal. Element a21 will become element a12 matrices and vice versa. As a result, from each initial matrices a new transposed matrix is obtained.
Add the given matrices, if they have the same dimension m x n. To do this, take the first matrices a11 and add it with the same element b11 second matrices. Write the result of the addition in a new one in the same position. Then add the elements a12 and b12 of both matrices. Thus, fill in all the rows and columns of the summation matrices.
Determine if given matrices agreed. To do this, compare the number of rows n in the first matrices and number of columns m second matrices. If they are equal, perform a matrix product. To do this, multiply each element of the first row in pairs. matrices to the corresponding element of the second column matrices. Then find the sum of these products. Thus, the first element of the resulting matrices g11 = a11*b11 + a12*b21 + a13*b31 + ... + a1m*bn1. Perform multiplication and addition of all products and fill in the resulting matrix G.
Find the determinant or determinant for each given matrices. For matrices of the second - with dimensions 2 by 2 - the determinant is found as the products of the elements of the main and secondary diagonals matrices. For 3D matrices determinant: D \u003d a11 * a22 * a33 + a13 * a21 * a32 + a12 * a23 * a31 - a21 * a12 * a33 - a13 * a22 * a31 - a11 * a32 * a23.
Sources:
- matrix how to solve
matrices are a set of rows and columns, at the intersection of which there are elements of the matrix. matrices are widely used to solve various equations. One of the basic algebraic operations on matrices is matrix addition. How to add matrices?
Instruction
You can add only one-dimensional matrices. If one has m rows and n columns, then the other matrix must also have m rows and n columns. Make sure that the stacking matrices are one-dimensional.
If the presented matrices are of the same size, that is, they allow the algebraic operation of addition, then when is a matrix of the same size. To make it, you need to add in pairs all the elements of two that are in the same places. Take the first matrix located in the first row and first column. Add it to the element of the second matrix, located in the same place. Enter the result in the element of the first row of the column of the total matrix. Do this for all elements.
The addition of three or more matrices is reduced to the addition of two matrices. For example, to find the sum of matrices A + B + C, first find the sum of matrices A and B, then add the resulting matrix to matrix C.
Related videos
Incomprehensible at first glance, the matrices are actually not so complicated. They find wide practical application in economics and accounting. Matrices look like tables, in each column and row containing a number, a function, or any other value. There are several types of matrices.
Instruction
In order to learn the matrix, get acquainted with its basic concepts. The defining elements of the matrix are its diagonals - and side. The main one starts with the element in the first row, first column and continues to the element of the last column, last row (that is, it goes from left to right). The side diagonal starts vice versa in the first row, but the last column and continues to the element that has the coordinates of the first column and the last row (goes from right to left).
In order to move on to the next definitions and algebraic operations with matrices, study the types of matrices. The simplest of them are square, unit, zero and inverse. In the same number of columns and rows. The transposed matrix, let's call it B, is obtained from matrix A by replacing columns with rows. In the unit, all elements of the main diagonal are ones, and the others are zeros. And in zero, even the elements of the diagonals are zero. The inverse matrix is the one on which the original matrix comes to the identity form.
Also, the matrix can be symmetrical about the main or side axes. That is, the element having coordinates a(1;2), where 1 is the row number and 2 is the column number, is equal to a(2;1). A(3;1)=A(1;3) and so on. Matched matrices are those where the number of columns of one is equal to the number of rows of the other (such matrices can be multiplied).
The main actions that can be performed with matrices are addition, multiplication and finding the determinant. If the matrices are the same size, that is, they have an equal number of rows and columns, then they can be added. It is necessary to add the elements that are in the same places in the matrices, that is, add a (m; n) with in (m; n), where m and n are the corresponding column and row coordinates. When adding matrices, the main rule of ordinary arithmetic addition applies - when the places of the terms are changed, the sum does not change. Thus, if instead of a simple element a
