A matrix is a rectangular table of numbers. Matrix concept
Actions on a matrix
1. Addition and subtraction of matrices:
Addition and subtraction of matrices- one of the simplest actions on them, because it is necessary to add or subtract the corresponding elements of two matrices. The main thing to remember is that only matrices can be added and subtracted same sizes, i.e. those that have the same number of rows and the same number of columns.
For example, let two matrices of equal size 2x3 be given, i.e. with two rows and three columns:
Sum of two matrices:
Difference of two matrices:
2. Multiplying a matrix by a number:
Multiplying a matrix by a number - the process of multiplying a number by each element of a matrix.
For example, let matrix A be given:
Let's multiply the number 3 by matrix A:
3. Multiplying two matrices:
Multiplying two matrices is possible only under the condition that the number of columns of the first matrix must be equal to the number of rows of the second. The new matrix, which will be obtained by multiplying matrices, will consist of a number of rows equal to the number of columns of the first matrix and a number of columns equal to the number of rows of the second matrix.
Suppose there are two matrices of dimensions 3x4 and 4x2, i.e. the first matrix has 3 rows and 4 columns, and the second matrix has 4 rows and 2 columns. Because the number of columns of the first matrix (4) is equal to the number of rows of the second matrix (4), then the matrices can be multiplied, the new matrix will have a size of 3x2, i.e. 3 rows and 2 columns.
You can imagine all this in the form of a diagram:
Once you have decided on the size of the new matrix that will be obtained by multiplying two matrices, you can begin to fill this matrix with elements. If you need to fill the first row of the first column of this matrix, then you need to multiply each element of the first row of the first matrix by each element of the first column of the second matrix, if we fill the second row of the first column, then we will take each element of the second row of the first matrix and multiply by the first column of the second matrices, etc.
Let's see what it looks like in the diagram:
Let's see what it looks like with an example:
Two matrices are given:
Let's find the product of these matrices:
4. Matrix division:
Matrix division- an action on matrices, which in this concept cannot be found in textbooks. But if there is a need to divide matrix A into matrix B, then in this case one of the properties of degrees is used:
According to this property, we divide matrix A by matrix B:
As a result, the problem of dividing matrices is reduced to multiplication inverse matrix matrix B to matrix A.
Inverse matrix
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 located in the last table under the matrix E of the original table.
Example 1
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 the inverse matrix A -1.
As a result of matrix multiplication, the identity matrix was obtained. Therefore, the calculations were performed correctly.
Answer: 
Determinants of matrices (Determinants) Determinants of matrices (Determinants)
Matrix determinants, method No. 1:
Determinant square matrix (det A) is a number that can be calculated from its elements matrices according to the formula:
Where M 1k - matrix determinant(determinant) obtained from the original matrices by crossing out the first row and the kth column. It should be noted that qualifiers have only square matrices, i.e. matrices in which the number of rows is equal to the number of columns. The first formula allows you to calculate matrix determinant according to the first line, the calculation formula is also valid determinant of the matrix for the first column: 
Generally speaking, matrix determinant can be calculated on any row or column matrices, i.e. the formula is correct:
Obviously, different matrices may have the same qualifiers. Determinant of the identity matrix equals 1. For the specified matrices And the number M 1k is called the additional minor of the element matrices a 1k. Thus, we can conclude that each element matrices has its own additional minor. Additional minors exist only in square matrices.
Additional minor of an arbitrary square element matrices a ij is equal to determinant of the matrix, obtained from the original matrices by crossing out the i-th row and j-th column.
Matrix determinants, method No. 2:
Matrix determinant first order, or determinant first order, element a 11 is called:
Matrix determinant second order, or determinant second order is a number that is calculated by the formula:
Matrix determinant third order, or determinant third order is a number that is calculated by the formula:
This number represents an algebraic sum consisting of six terms. Each term contains exactly one element from each row and each column matrices. Each term consists of the product of three factors.
Signs with which members determinant of the matrix included in the formula finding the determinant of the matrix third order can be determined using the given scheme, which is called the rule of triangles or Sarrus's rule. The first three terms are taken with a plus sign and determined from the left figure, and the next three terms are taken with a minus sign and determined from the right figure.
Comment:
Calculation matrix determinants fourth and higher order leads to large calculations because:
For of the first order we find one term consisting of one factor;
For finding the determinant of the matrix of the second order, you need to calculate an algebraic sum of two terms, where each term consists of the product of two factors;
For finding the determinant of the matrix third order, you need to calculate an algebraic sum of six terms, where each term consists of the product of three factors;
For finding the determinant of the matrix fourth order, you need to calculate an algebraic sum of twenty-four terms, where each term consists of the product of four factors, etc.
Determine the number of terms to find determinant of the matrix, in an algebraic sum, you can calculate the factorial: 1!=1 2!=1×2=2 3!=1×2×3=6 4!=1×2×3×4=24 5! = 1 × 2 × 3 × 4 × 5 = 120 ...
Matrices in mathematics are one of the most important objects of practical 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 direction.
Phone books of any size and with any amount of subscriber data are nothing more than matrices. Such matrices look approximately like this:
It is clear that we all use such matrices almost every day. These matrices come with a different number of rows (they vary like a directory issued by a telephone company, which can have thousands, hundreds of thousands and even millions of rows and the new one you just started notebook, in which there are less than ten lines) and columns (a directory of officials of some organization, which may contain columns such as position and office number and your same notebook, where there may not be any data except the name, and thus Thus, it has only two columns - name and phone number).
All sorts of matrices can be added and multiplied, as well as other operations can be performed on them, but there is no need to add and multiply telephone directories, there is no benefit from this, besides, you can move your mind.
But many matrices can and should be added and multiplied and thus solve various pressing problems. Below are examples of such matrices.
Matrices in which the columns are the production of units of a particular type of product, and the rows are the years in which the production of this product is recorded:
You can add matrices of this type, which take into account the output of similar products by different 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:
The last two types of matrices can be multiplied, and the result is a row matrix containing the cost of all types of products by year.
Matrices, basic definitions
A rectangular table consisting of numbers arranged in m lines and n columns is called mn-matrix (or just matrix ) and is 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 the element is located; 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 .
Determinant of a square matrix A is a determinant whose elements are the elements of a matrix A. It is indicated by the symbol | A|.
The square matrix is called not special (or non-degenerate , non-singular ), if its determinant is not 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 corresponding elements match.
The matrix is called null , if all its elements are equal to zero. Zero matrix we will denote by the symbol 0 or .
For example,
Matrix-row (or lowercase ) is called 1 n-matrix, and matrix-column (or columnar ) – m 1-matrix.
Matrix A", which is obtained from the matrix A swapping rows and columns in it is called transposed relative to the matrix A. Thus, for matrix (1) the transposed matrix is
Matrix transition operation A" transposed with respect to the matrix A, is called matrix transposition A. For mn-matrix transposed is nm-matrix.
The matrix transposed with respect to the matrix is A, that is
(A")" = A .
Example 1. Find matrix A" , transposed with respect to the matrix
and find out whether 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 off the main diagonal are equal to zero is called diagonal . Not all diagonal elements of a diagonal matrix are necessarily non-zero. Among them there may be equal to zero.
A square matrix in which the elements on the main diagonal are equal to the same number, non-zero, 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 third-order identity matrix is the matrix
Example 2. Given matrices:
Solution. Let us calculate the determinants of these matrices. Using the triangle rule, we find
Matrix determinant B let's calculate using 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 the identity matrix of any order is obviously equal to one.
Solve the matrix problem yourself, and then look at the solution
Example 3. Given matrices
,
,
Determine which of them are non-singular (non-degenerate, non-singular).
Application of matrices in mathematical and economic modeling
Structured data about a particular object is simply and conveniently recorded in the form of matrices. Matrix models are created not only to store this structured data, but also to solve various problems with these data means linear algebra.
Thus, a well-known matrix model of the economy is the input-output model, introduced by the American economist of Russian origin Vasily Leontiev. This model is based on the assumption that the entire production sector of the economy is divided into n clean industries. Each industry produces only one type of product, and different industries produce different products. Due to this division of labor between industries, there are inter-industry connections, the meaning of which is that part of the production of each industry is transferred to other industries as a production resource.
Product volume i-th industry (measured by a specific unit of measurement), which was produced during the reporting period, is denoted by and is called full output i-th industry. Issues can be conveniently placed in n-component row of the matrix.
Number of units i-industry that needs to be spent j-industry for the production of a unit of its output is designated and called the direct cost coefficient.
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. That is, the entries below mean the same matrix:
$$ \left(\begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right);\;\; \left[ \begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right]; \;\; \left \Vert \begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right \Vert $$
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. You can write it in a little more detail:
$$ A_(m\times n)=(a_(ij)) $$
where the notation $(a_(ij))$ denotes the elements of the matrix $A$. In its fully 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.
The element $a_(12)$ is at the intersection of the first row and the 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 view 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 the matrix $A_(m\times n)$ have the following form:
Then this matrix is called trapezoidal. It may not contain zero rows, but if they exist, they are located at the bottom of the matrix. In a more general form, a trapezoidal matrix can be written as follows:
Again, trailing null lines are not required. Those. Formally, we can distinguish the following conditions for a trapezoidal matrix:
- All elements below the main diagonal are zero.
- All elements from $a_(11)$ to $a_(rr)$ lying on the main diagonal are not equal to zero: $a_(11)\neq 0, \; a_(22)\neq 0, \ldots, a_(rr)\neq 0$.
- Either all elements of the last $m-r$ rows are zero, or $m=r$ (i.e. there are no zero rows at all).
Examples of trapezoidal matrices:
Let's move on to the next definition. The matrix $A_(m\times n)$ is called stepped, if it satisfies the following conditions:
For example, step matrices would be:
For comparison, the matrix $\left(\begin(array) (cccc) 2 & -2 & 0 & 1\\0 & 0 & 8 & 7\\0 & 0 & 4 & -7\\0 & 0 & 0 & 0 \end(array)\right)$ is not echelon because the third row has the same zero part as the second row. That is, the principle “the lower the line, the larger the zero part” is violated. Let me add that there is a trapezoidal matrix special case step matrix.
Let's move on to the next definition. 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.
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 a rectangular matrix of 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 of the 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.
Any matrix can be multiplied by an arbitrary number. 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 be equal to the sum of the 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 of linear algebra. Once upon a time people came up with linear equations, and behind them we 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 a numerical characteristic of a 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, calculating 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 a matrix system of equations, or, on the contrary, you may encounter 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.
Matrices, get acquainted with its basic concepts. The defining elements of a matrix are its diagonals and its side diagonals. Home starts with the element in the first row, first column and continues to the element in the last column, last row (that is, it goes from left to right). The side diagonal begins on the contrary in the first row, but last column and continues to the element having the coordinates of the first column and last row (going from right to left).
In order to go to following definitions and algebraic operations with matrices, study the types of matrices. The simplest ones are square, unit, zero and inverse. The number of columns and rows matches. The transposed matrix, let's call it B, is obtained from matrix A by replacing the columns with rows. In unit, all the 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 secondary axes. That is, an 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 another (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 elements that are in the same places in the matrices, that is, add a (m; n) with c in (m; n), where m and n are the corresponding coordinates of the column and row. 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 there is an expression a + b, then it can be added to an element c of another commensurate matrix according to the rules a + (b + c) = (a + b) + c.
You can multiply the matched matrices given above. This produces a matrix where each element is the sum of the pairwise multiplied elements of a row of matrix A and a column of matrix B. When multiplying, the order of actions is very important. m*n is not equal to n*m.
Also one of the main actions is finding. It is also called a determinant and is designated as follows: det. This value is determined modulo, that is, it is never negative. The easiest way to find the determinant is a 2x2 square matrix. To do this, you need to multiply the elements of the main diagonal and subtract from them the multiplied elements of the secondary diagonal.