Product of a matrix by a number online. Basic operations on matrices (addition, multiplication, transposition) and their properties. Properties of multiplying a matrix by a number

This topic will cover such operations as addition and subtraction of matrices, multiplication of a matrix by a number, multiplication of a matrix by a matrix, and transposition of a matrix. All symbols used on this page are taken from the previous topic.

Addition and subtraction of matrices.

The sum $ A + B $ of matrices $ A_ (m \ times n) = (a_ (ij)) $ and $ B_ (m \ times n) = (b_ (ij)) $ is called the matrix $ C_ (m \ times n) = (c_ (ij)) $, where $ c_ (ij) = a_ (ij) + b_ (ij) $ for all $ i = \ overline (1, m) $ and $ j = \ overline (1, n) $.

A similar definition is introduced for the difference of matrices:

The difference $ AB $ of matrices $ A_ (m \ times n) = (a_ (ij)) $ and $ B_ (m \ times n) = (b_ (ij)) $ is the matrix $ C_ (m \ times n) = ( c_ (ij)) $, where $ c_ (ij) = a_ (ij) -b_ (ij) $ for all $ i = \ overline (1, m) $ and $ j = \ overline (1, n) $.

Explanation of the entry $ i = \ overline (1, m) $: show \ hide

The notation "$ i = \ overline (1, m) $" means that the $ i $ parameter ranges from 1 to m. For example, the record $ i = \ overline (1,5) $ says that the parameter $ i $ takes the values 1, 2, 3, 4, 5.

It should be noted that addition and subtraction operations are defined only for matrices of the same size. In general, addition and subtraction of matrices are intuitively clear operations, because they mean, in fact, just the addition or subtraction of the corresponding elements.

Example # 1

Three matrices are given:

$$ A = \ left (\ begin (array) (ccc) -1 & -2 & 1 \\ 5 & 9 & -8 \ end (array) \ right) \; \; B = \ left (\ begin (array) (ccc) 10 & -25 & 98 \\ 3 & 0 & -14 \ end (array) \ right); \; \; F = \ left (\ begin (array) (cc) 1 & 0 \\ -5 & 4 \ end (array) \ right). $$

Can you find the $ A + F $ matrix? Find matrices $ C $ and $ D $ if $ C = A + B $ and $ D = A-B $.

The $ A $ matrix contains 2 rows and 3 columns (in other words, the size of the $ A $ matrix is $ 2 \ times 3 $), and the $ F $ matrix contains 2 rows and 2 columns. The sizes of the matrix $ A $ and $ F $ do not coincide, so we cannot add them, i.e. the $ A + F $ operation for these matrices is undefined.

The sizes of the matrices $ A $ and $ B $ are the same, i.e. the matrix data contains an equal number of rows and columns, so the addition operation is applicable to them.

$$ C = A + B = \ left (\ begin (array) (ccc) -1 & -2 & 1 \\ 5 & 9 & -8 \ end (array) \ right) + \ left (\ begin (array ) (ccc) 10 & -25 & 98 \\ 3 & 0 & -14 \ end (array) \ right) = \\ = \ left (\ begin (array) (ccc) -1 + 10 & -2+ ( -25) & 1 + 98 \\ 5 + 3 & 9 + 0 & -8 + (- 14) \ end (array) \ right) = \ left (\ begin (array) (ccc) 9 & -27 & 99 \\ 8 & 9 & -22 \ end (array) \ right) $$

Find the matrix $ D = A-B $:

$$ D = AB = \ left (\ begin (array) (ccc) -1 & -2 & 1 \\ 5 & 9 & -8 \ end (array) \ right) - \ left (\ begin (array) ( ccc) 10 & -25 & 98 \\ 3 & 0 & -14 \ end (array) \ right) = \\ = \ left (\ begin (array) (ccc) -1-10 & -2 - (- 25 ) & 1-98 \\ 5-3 & 9-0 & -8 - (- 14) \ end (array) \ right) = \ left (\ begin (array) (ccc) -11 & 23 & -97 \ \ 2 & 9 & 6 \ end (array) \ right) $$

Answer: $ C = \ left (\ begin (array) (ccc) 9 & -27 & 99 \\ 8 & 9 & -22 \ end (array) \ right) $, $ D = \ left (\ begin (array) (ccc) -11 & 23 & -97 \\ 2 & 9 & 6 \ end (array) \ right) $.

Multiplication of a matrix by a number.

The product of the matrix $ A_ (m \ times n) = (a_ (ij)) $ by the number $ \ alpha $ is the matrix $ B_ (m \ times n) = (b_ (ij)) $, where $ b_ (ij) = \ alpha \ cdot a_ (ij) $ for all $ i = \ overline (1, m) $ and $ j = \ overline (1, n) $.

Simply put, multiplying a matrix by a certain number means multiplying each element of a given matrix by that number.

Example No. 2

The matrix is given: $ A = \ left (\ begin (array) (ccc) -1 & -2 & 7 \\ 4 & 9 & 0 \ end (array) \ right) $. Find the matrices $ 3 \ cdot A $, $ -5 \ cdot A $ and $ -A $.

$$ 3 \ cdot A = 3 \ cdot \ left (\ begin (array) (ccc) -1 & -2 & 7 \\ 4 & 9 & 0 \ end (array) \ right) = \ left (\ begin ( array) (ccc) 3 \ cdot (-1) & 3 \ cdot (-2) & 3 \ cdot 7 \\ 3 \ cdot 4 & 3 \ cdot 9 & 3 \ cdot 0 \ end (array) \ right) = \ left (\ begin (array) (ccc) -3 & -6 & 21 \\ 12 & 27 & 0 \ end (array) \ right). \\ -5 \ cdot A = -5 \ cdot \ left (\ begin (array) (ccc) -1 & -2 & 7 \\ 4 & 9 & 0 \ end (array) \ right) = \ left (\ begin (array) (ccc) -5 \ cdot (-1) & - 5 \ cdot (-2) & -5 \ cdot 7 \\ -5 \ cdot 4 & -5 \ cdot 9 & -5 \ cdot 0 \ end (array) \ right) = \ left (\ begin (array) ( ccc) 5 & 10 & -35 \\ -20 & -45 & 0 \ end (array) \ right). $$

The $ -A $ notation is a shorthand for $ -1 \ cdot A $. That is, to find $ -A $, you need to multiply all the elements of the $ A $ matrix by (-1). In fact, this means that the sign of all elements of the matrix $ A $ will change to the opposite:

$$ -A = -1 \ cdot A = -1 \ cdot \ left (\ begin (array) (ccc) -1 & -2 & 7 \\ 4 & 9 & 0 \ end (array) \ right) = \ left (\ begin (array) (ccc) 1 & 2 & -7 \\ -4 & -9 & 0 \ end (array) \ right) $$

Answer: $ 3 \ cdot A = \ left (\ begin (array) (ccc) -3 & -6 & 21 \\ 12 & 27 & 0 \ end (array) \ right); \; -5 \ cdot A = \ left (\ begin (array) (ccc) 5 & 10 & -35 \\ -20 & -45 & 0 \ end (array) \ right); \; -A = \ left (\ begin (array) (ccc) 1 & 2 & -7 \\ -4 & -9 & 0 \ end (array) \ right) $.

Product of two matrices.

The definition of this operation is cumbersome and, at first glance, incomprehensible. Therefore, first I will indicate a general definition, and then we will analyze in detail what it means and how to work with it.

The matrix $ C_ (m \ times k) = (c_ ( ij)) $, for which each element of $ c_ (ij) $ is equal to the sum of the products of the corresponding elements of the i-th row of the matrix $ A $ by the elements of the j-th column of the matrix $ B $: $$ c_ (ij) = \ sum \ limits_ (p = 1) ^ (n) a_ (ip) b_ (pj), \; \; i = \ overline (1, m), j = \ overline (1, n). $$

Let us analyze step-by-step matrix multiplication using an example. However, you should immediately pay attention to the fact that not all matrices can be multiplied. If we want to multiply the $ A $ matrix by the $ B $ matrix, then first we need to make sure that the number of columns of the $ A $ matrix is equal to the number of rows of the $ B $ matrix (such matrices are often called agreed). For example, the matrix $ A_ (5 \ times 4) $ (the matrix contains 5 rows and 4 columns) cannot be multiplied by the matrix $ F_ (9 \ times 8) $ (9 rows and 8 columns), since the number of columns of the matrix $ A $ is not equal to the number of rows in the $ F $ matrix, i.e. $ 4 \ neq 9 $. But you can multiply the matrix $ A_ (5 \ times 4) $ by the matrix $ B_ (4 \ times 9) $, since the number of columns in the matrix $ A $ is equal to the number of rows in the matrix $ B $. In this case, the result of multiplying the matrices $ A_ (5 \ times 4) $ and $ B_ (4 \ times 9) $ will be the matrix $ C_ (5 \ times 9) $, containing 5 rows and 9 columns:

Example No. 3

Matrices are given: $ A = \ left (\ begin (array) (cccc) -1 & 2 & -3 & 0 \\ 5 & 4 & -2 & 1 \\ -8 & 11 & -10 & -5 \ end (array) \ right) $ and $ B = \ left (\ begin (array) (cc) -9 & 3 \\ 6 & 20 \\ 7 & 0 \\ 12 & -4 \ end (array) \ right) $. Find the matrix $ C = A \ cdot B $.

To begin with, let's immediately determine the size of the $ C $ matrix. Since $ A $ is $ 3 \ times 4 $ and $ B $ is $ 4 \ times 2 $, the size of $ C $ is $ 3 \ times 2 $:

So, as a result of the product of the matrices $ A $ and $ B $, we should get the matrix $ C $, consisting of three rows and two columns: $ C = \ left (\ begin (array) (cc) c_ (11) & c_ ( 12) \\ c_ (21) & c_ (22) \\ c_ (31) & c_ (32) \ end (array) \ right) $. If the designations of the elements raise questions, then you can look at the previous topic: "Matrices. Types of matrices. Basic terms", at the beginning of which the designation of the matrix elements is explained. Our goal is to find the values of all elements of the matrix $ C $.

Let's start with $ c_ (11) $. To get the element $ c_ (11) $, you need to find the sum of the products of the elements of the first row of the matrix $ A $ and the first column of the matrix $ B $:

To find the element $ c_ (11) $ itself, you need to multiply the elements of the first row of the matrix $ A $ by the corresponding elements of the first column of the matrix $ B $, i.e. the first element to the first, the second to the second, the third to the third, the fourth to the fourth. We summarize the results obtained:

$$ c_ (11) = - 1 \ cdot (-9) +2 \ cdot 6 + (- 3) \ cdot 7 + 0 \ cdot 12 = 0. $$

Let's continue the solution and find $ c_ (12) $. To do this, you have to multiply the elements of the first row of the matrix $ A $ and the second column of the matrix $ B $:

Similar to the previous one, we have:

$$ c_ (12) = - 1 \ cdot 3 + 2 \ cdot 20 + (- 3) \ cdot 0 + 0 \ cdot (-4) = 37. $$

All elements of the first row of $ C $ are found. Move on to the second line, which starts with $ c_ (21) $. To find it, you have to multiply the elements of the second row of the matrix $ A $ and the first column of the matrix $ B $:

$$ c_ (21) = 5 \ cdot (-9) +4 \ cdot 6 + (- 2) \ cdot 7 + 1 \ cdot 12 = -23. $$

The next element $ c_ (22) $ is found by multiplying the elements of the second row of the matrix $ A $ by the corresponding elements of the second column of the matrix $ B $:

$$ c_ (22) = 5 \ cdot 3 + 4 \ cdot 20 + (- 2) \ cdot 0 + 1 \ cdot (-4) = 91. $$

To find $ c_ (31) $ we multiply the elements of the third row of the matrix $ A $ by the elements of the first column of the matrix $ B $:

$$ c_ (31) = - 8 \ cdot (-9) +11 \ cdot 6 + (- 10) \ cdot 7 + (-5) \ cdot 12 = 8. $$

And, finally, to find the element $ c_ (32) $, you will have to multiply the elements of the third row of the matrix $ A $ by the corresponding elements of the second column of the matrix $ B $:

$$ c_ (32) = - 8 \ cdot 3 + 11 \ cdot 20 + (- 10) \ cdot 0 + (-5) \ cdot (-4) = 216. $$

All the elements of the matrix $ C $ are found, it remains only to write that $ C = \ left (\ begin (array) (cc) 0 & 37 \\ -23 & 91 \\ 8 & 216 \ end (array) \ right) $ ... Or, to write in full:

$$ C = A \ cdot B = \ left (\ begin (array) (cccc) -1 & 2 & -3 & 0 \\ 5 & 4 & -2 & 1 \\ -8 & 11 & -10 & - 5 \ end (array) \ right) \ cdot \ left (\ begin (array) (cc) -9 & 3 \\ 6 & 20 \\ 7 & 0 \\ 12 & -4 \ end (array) \ right) = \ left (\ begin (array) (cc) 0 & 37 \\ -23 & 91 \\ 8 & 216 \ end (array) \ right). $$

Answer: $ C = \ left (\ begin (array) (cc) 0 & 37 \\ -23 & 91 \\ 8 & 216 \ end (array) \ right) $.

By the way, there is often no reason to describe in detail the finding of each element of the result matrix. For matrices whose size is small, you can do the following:

$$ \ left (\ begin (array) (cc) 6 & 3 \\ -17 & -2 \ end (array) \ right) \ cdot \ left (\ begin (array) (cc) 4 & 9 \\ - 6 & 90 \ end (array) \ right) = \ left (\ begin (array) (cc) 6 \ cdot (4) +3 \ cdot (-6) & 6 \ cdot (9) +3 \ cdot (90 ) \\ -17 \ cdot (4) + (- 2) \ cdot (-6) & -17 \ cdot (9) + (- 2) \ cdot (90) \ end (array) \ right) = \ left (\ begin (array) (cc) 6 & 324 \\ -56 & -333 \ end (array) \ right) $$

It is also worth noting that matrix multiplication is non-commutative. This means that in general $ A \ cdot B \ neq B \ cdot A $. Only for some types of matrices, which are called permutation(or commuting), the equality $ A \ cdot B = B \ cdot A $ is true. It is on the basis of the non-commutativity of multiplication that it is required to indicate exactly how we multiply the expression by this or that matrix: to the right or to the left. For example, the phrase "multiply both sides of the equality $ 3E-F = Y $ by the matrix $ A $ on the right" means that we need to obtain the following equality: $ (3E-F) \ cdot A = Y \ cdot A $.

Transposed with respect to the matrix $ A_ (m \ times n) = (a_ (ij)) $ is called the matrix $ A_ (n \ times m) ^ (T) = (a_ (ij) ^ (T)) $, for elements which $ a_ (ij) ^ (T) = a_ (ji) $.

Simply put, in order to obtain the transposed matrix $ A ^ T $, you need to replace the columns in the original matrix $ A $ with the corresponding rows according to the following principle: if the first row was, the first column will become; there was a second line - there will be a second column; there was a third line - there will be a third column and so on. For example, let's find the transposed matrix to the matrix $ A_ (3 \ times 5) $:

Accordingly, if the original matrix was $ 3 \ times 5 $, then the transposed matrix is $ 5 \ times 3 $.

Some properties of operations on matrices.

It is assumed here that $ \ alpha $, $ \ beta $ are some numbers, and $ A $, $ B $, $ C $ are matrices. For the first four properties, I indicated the names, the rest can be named by analogy with the first four.

Lecture number 1

MATRIX

Definition and types of matrices

Definition 1.1.Matrix size T NS called a rectangular table of numbers (or other objects) containing m lines and n columns.

Matrices are denoted by uppercase (capital) letters of the Latin alphabet, for example, A, B, C, ... The numbers (or other objects) that make up the matrix are called elements matrices. The elements of the matrix can be functions. To designate the elements of the matrix, lowercase letters of the Latin alphabet with double indexing are used: аij, where the first index i(read - and) - line number, second index j(read - zhi) – column number.

Definition 1.2. The matrix is called square n- th order if the number of its rows is equal to the number of columns and is equal to the same number NS

For a square matrix, the concepts are introduced main and secondary diagonals.

Definition 1.3.Main diagonal a square matrix consists of elements with the same indices, i.e. These are the elements: a 11, a 22, ...

Definition 1.4. diagonal if all elements, except for the elements of the main diagonal, are equal to zero

Definition 1.5. The square matrix is called triangular if all its elements located below (or above) the main diagonal are equal to zero.

Definition 1.6. Square matrix NS- th order, in which all elements of the main diagonal are equal to one, and the rest are equal to zero, is called single matrix n-th order, and it is denoted by the letter E.

Definition 1.7. A matrix of any size is called null, or null matrix, if all its elements are equal to zero.

Definition 1.8. A one-row matrix is called matrix-row.

Definition 1.9. A single column matrix is called column matrix.

A = (a 11 a 12 ... a 1n) - row matrix;

Definition 1.10. Two matrices A and V of the same size are called equal, if all the corresponding elements of these matrices are equal to each other, i.e. aij = bij for any i= 1, 2, ..., T; j = 1, 2,…, n.

Matrix operations

A number of operations can be performed on matrices, as well as on numbers. The main operations on matrices are addition (subtraction) of matrices, multiplication of a matrix by a number, and matrix multiplication. These operations are similar to operations on numbers. A specific operation is matrix transposition.

Multiplying a Matrix by a Number

Definition 1.11.The product of the matrix A by the numberλ is called the matrix B = A, whose elements are obtained by multiplying the elements of the matrix A by the number λ .

Example 1.1. Find the product of a matrix A = to the number 5.

Solution... .◄ 5A =

The rule of multiplying a matrix by a number: to multiply a matrix by a number, all the elements of the matrix must be multiplied by this number.

Consequence.

1. The common factor of all matrix elements can be taken out of the matrix sign.

2. The product of the matrix A there is a zero matrix for the number 0: A· 0 = 0 .

Matrix addition

Definition 1.12.The sum of two matrices A and B the same size t n called the matrix WITH= A+ V, whose elements are obtained by adding the corresponding elements of the matrix A and matrices V, i.e. cij = aij + bij for i = 1, 2, ..., m; j= 1, 2, ..., n(i.e., matrices are added element by element).

Consequence. Matrix sum A with a zero matrix is equal to the original matrix: A + O = A.

1.2.3. Subtraction of matrices

Difference of two matrices the same size is determined through the previous operations: A - B = A + (- 1)V.

Definition 1.13. Matrix –A = (- 1)A called opposite matrix A.

Consequence. The sum of the opposite matrices is equal to the zero matrix : A + (–A) = O.

Matrix multiplication

Definition 1.14.Multiplication of matrix A by matrix B it is determined when the number of columns of the first matrix is equal to the number of rows of the second matrix. Then product of matrices such a matrix is called , each element of which cij is equal to the sum of the products of the elements i-th row of the matrix A on the corresponding elements j th column of the matrix B.

Example 1.4. Calculate the product of matrices А В, where

A =

Example 1.5. Find Matrix Products AB and VA, where

Remarks. From Examples 1.4–1.5 it follows that the operation of matrix multiplication has some differences from multiplication of numbers:

1) if the product of matrices AB exists, then after rearranging the factors in places, the product of matrices VA may not exist. Indeed, in Example 1.4, the product of matrices AB exists, but the product BA does not exist;

2) even if the works AB and VA exist, then the result of the product can be matrices of different sizes. In the case when both works AB and VA both exist - matrices of the same size (this is possible only when multiplying square matrices of the same order), then the commutative (transposable) law of multiplication still does not hold, those. A B B A, as in example 1.5;

3) however, if we multiply the square matrix A on the identity matrix E of the same order, then AE = EA = A.

Thus, the identity matrix in matrix multiplication plays the same role as the number 1 in multiplication of numbers;

4) the product of two nonzero matrices can be equal to the zero matrix, that is, from the fact that A B= 0, it does not follow that A = 0 or B = 0.

1st year, higher mathematics, we study matrices and basic actions on them. Here we systematize the basic operations that can be performed with matrices. Where to start acquaintance with matrices? Of course, from the simplest - definitions, basic concepts and simplest operations. We assure you that matrices will be understood by everyone who devotes at least a little time to them!

Definition of a matrix

Matrix Is a rectangular table of elements. Well, if in simple terms - a table of numbers.

Typically, matrices are indicated by uppercase Latin letters. For example, the matrix A , matrix B etc. Matrices can be of 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 - the number of lines, and n - the number of columns.

Elements 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

We warn you right away that you can only add matrices of the same size. The result is a matrix of the same size. Adding (or subtracting) matrices is easy - just add their respective elements ... Let's give an example. Let's add two matrices A and B, 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 among themselves. 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, standing in the i-th row and j-th column, will be equal to the sum of the products of the corresponding elements in the i-th row of 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 matrices:

Matrix transpose operation

Matrix transpose is an operation where the corresponding rows and columns are swapped. For example, let's transpose the matrix A from the first example:

Determinant of a matrix

Determinant, but determinant is one of the basic concepts of linear algebra. Once upon a time people invented linear equations, and behind them they had to invent a determinant. As a result, you have to deal with all this, so, the last spurt!

A 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 the first order, that is, consisting of one element, is equal to this element.

What if the matrix is three by three? Here it is already more complicated, but you can cope.

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 side diagonal and the product of the elements lying on the triangles with the face of the parallel side diagonal are subtracted.

Fortunately, it is rarely necessary to calculate determinants of large matrices in practice.

Here we have covered the basic operations on matrices. Of course, in real life, you may never come across even a hint of a matrix system of equations, or vice versa - to face much more difficult cases when you really have to break your head. It is for such cases that there is a professional student service. Ask for help, get a high-quality and detailed solution, enjoy your academic success and free time.

In order to multiply the matrix A by an arbitrary number α, you need the elements of the matrix A multiply by the number α, i.e. the product of a matrix by a number will be as follows:

Example 1. Find Matrix 3 A for matrix

Solution. In accordance with the definition, we multiply the elements of the matrix A by 3 and we get

It was a very simple example of multiplying a matrix by a number with integers. There are also simple examples ahead, but already such where among the factors and elements of matrices there are fractions, variables (letter designations), because the laws of multiplication are valid not only for integers, so it is never harmful to repeat them.

Example 2. A by the number α if
, .

A by α, not forgetting that when multiplying fractions, the numerator of the first fraction is multiplied by the numerator of the first fraction and the product is written into the numerator, and the denominator of the first fraction is multiplied by the denominator of the second fraction and the product is written into the denominator. When receiving the second element of the first row of the new matrix, the resulting fraction was reduced by 2, this must be done. We get

Example 3. Perform matrix multiplication operation A by the number α if
, .

Solution. Multiply the elements of the matrix A by α, without getting confused in the letter notation, not forgetting to leave a minus in front of the second element of the second row of the new matrix, and remembering that the result of multiplying a number by its inverse number is one (the first element of the third row). We get

Example 4. Perform matrix multiplication operation A by the number α if
, .

Solution. Remember that when you multiply a number in a power by a number in a power, the exponents are added. We get

This example, among other things, clearly demonstrates that the operations of multiplying a matrix by a number can be read (and written) in reverse order, and this is called putting a constant factor in front of the matrix.

Combined with addition and subtraction of matrices the operation of multiplying a matrix by a number can form various matrix expressions, for example, 5 A − 3B , 4A + 2B .

Properties of multiplying a matrix by a number

(here A, B - matrices, - numbers, 1 - number one)

Properties (1) and (2) connect the multiplication of a matrix by a number with the addition of matrices. There is also a very important connection between multiplying a matrix by a number and multiplying the matrices themselves:

that is, if in the product of matrices one of the factors is multiplied by a number, then the entire product will be multiplied by a number.

This methodological guide will help you learn how to perform operations with matrices: addition (subtraction) of matrices, transposition of a matrix, multiplication of matrices, finding the inverse matrix. All material is presented in a simple and accessible form, relevant examples are given, so even an unprepared person can learn how to perform actions with matrices. For self-checking and self-checking, you can download a matrix calculator for free >>>.

I will try to minimize theoretical calculations, in some places explanations are possible "on the fingers" and the use of unscientific terms. Lovers of solid theory, please do not criticize, our task is learn to perform actions with matrices.

For SUPER-FAST preparation on the topic (who is "on fire") there is an intensive pdf-course Matrix, determinant and test!

Matrix is a rectangular table of any elements... As elements we will consider numbers, that is, numeric matrices. ELEMENT Is a term. It is advisable to remember the term, it will be often encountered, it is no coincidence that I used bold type to highlight it.

Designation: matrices are usually denoted by uppercase Latin letters

Example: Consider a two-by-three matrix:

This matrix consists of six elements:

All numbers (elements) inside the matrix exist by themselves, that is, there is no question of any subtraction:

It's just a table (set) of numbers!

We will also agree do not rearrange numbers, unless otherwise stated in the explanations. Each number has its own location and cannot be shuffled!

The matrix in question has two rows:

and three columns:

STANDARD: when talking about the size of the matrix, then at first indicate the number of rows, and only then - the number of columns. We've just taken apart a two-by-three matrix.

If the number of rows and columns of the matrix is the same, then the matrix is called square, for example: - a three-by-three matrix.

If the matrix has one column or one row, then such matrices are also called vectors.

In fact, we know the concept of a matrix since school, consider, for example, a point with coordinates "x" and "game":. Essentially, the coordinates of a point are written in a one-by-two matrix. By the way, here's an example for you why the order of numbers matters: and are two completely different points on the plane.

Now let's go directly to the study actions with matrices:

1) First action. Removing the minus from the matrix (adding the minus to the matrix).

Back to our matrix ... As you may have noticed, there are too many negative numbers in this matrix. This is very inconvenient from the point of view of performing various actions with the matrix, it is inconvenient to write so many minuses, and it just looks ugly in design.

Move the minus outside the matrix by changing the sign of EACH matrix element:

At zero, as you understand, the sign does not change, zero - it is zero in Africa.

Reverse example: ... It looks ugly.

Let's add a minus to the matrix by changing the sign of EACH matrix element:

Well, it turned out much nicer. And, most importantly, it will be EASIER to perform any actions with the matrix. Because there is such a mathematical folk omen: the more cons, the more confusion and mistakes.

2) Second action. Multiplying a Matrix by a Number.

Example:

It's simple, in order to multiply a matrix by a number, you need each the element of the matrix is multiplied by the given number. In this case, the top three.

Another useful example:

- matrix multiplication by a fraction

Let's look at what to do first. NO NEED:

It is NOT NECESSARY to enter a fraction into the matrix, firstly, it only complicates further actions with the matrix, and secondly, it makes it difficult for the teacher to check the solution (especially if - the final answer of the task).

And especially, NO NEED divide each element of the matrix by minus seven:

From article Math for dummies or where to start, we remember that decimal fractions with a comma in higher mathematics are tried in every possible way to avoid.

The only thing that desirable to do in this example is to introduce a minus into the matrix:

But if ALL matrix elements were divisible by 7 without a remainder, then it would be possible (and necessary!) to divide.

Example:

In this case, it is possible and NECESSARY multiply all the elements of the matrix by, since all the numbers in the matrix are divisible by 2 without a remainder.

Note: in the theory of higher mathematics there is no school concept of "division". Instead of the phrase "divide this by this" you can always say "multiply this by a fraction." That is, division is a special case of multiplication.

3) Third action. Matrix transpose.

In order to transpose a matrix, you need to write its rows into the columns of the transposed matrix.

Example:

Transpose Matrix

There is only one line here and, according to the rule, it must be written to a column:

- transposed matrix.

A transposed matrix is usually indicated by a superscript or a dash at the top right.

Step by step example:

Transpose Matrix

First, we rewrite the first row to the first column:

Then we rewrite the second line into the second column:

Finally, we rewrite the third line into the third column:

Ready. Roughly speaking, to transpose means to turn the matrix to one side.

4) Action four. Sum (difference) of matrices.

The sum of the matrices is a simple operation.
NOT ALL DIES CAN FOLD. To perform the addition (subtraction) of matrices, it is necessary that they be the same SIZE.

For example, if a two-by-two matrix is given, then it can only be added with a two-by-two matrix and no other!

Example:

Add matrices and

In order to add matrices, it is necessary to add their corresponding elements:

For the difference of matrices, the rule is similar, it is necessary to find the difference of the corresponding elements.

Example:

Find the difference of matrices ,

And how to solve this example easier so as not to get confused? It is advisable to get rid of unnecessary minuses, for this we add a minus to the matrix:

Note: in the theory of higher mathematics there is no school concept of "subtraction". Instead of saying "subtract this from this" you can always say "add a negative number to this." That is, subtraction is a special case of addition.

5) Action five. Matrix multiplication.

What matrices can be multiplied?

In order for the matrix to be multiplied by the matrix, you need so that the number of columns of the matrix is equal to the number of rows of the matrix.

Example:
Is it possible to multiply a matrix by a matrix?

This means that you can multiply these matrices.

But if the matrices are rearranged, then, in this case, multiplication is already impossible!

Therefore, multiplication is not possible:

It is not so rare that tasks with a trick are encountered when a student is asked to multiply matrices, the multiplication of which is obviously impossible.

It should be noted that in some cases it is possible to multiply matrices both ways.
For example, for matrices, and both multiplication and multiplication are possible