This guide will help you learn how to matrix operations: addition (subtraction) of matrices, transposition of a matrix, multiplication of matrices, finding the inverse of a 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-control and self-test, you can download a matrix calculator for free >>>.
I will try to minimize theoretical calculations, in some places explanations “on the fingers” and the use of unscientific terms are possible. Lovers of solid theory, please do not engage in criticism, our task is learn how to work with matrices.
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A matrix is a rectangular table of some elements. As elements we will consider numbers, that is, numerical matrices. ELEMENT is a term. It is desirable to remember the term, it will often occur, it is no coincidence that I used bold to highlight it.
Designation: matrices are usually denoted by capital Latin letters
Example: Consider a two-by-three matrix:
This matrix consists of six elements:
All numbers (elements) inside the matrix exist on their own, that is, there is no question of any subtraction: 
It's just a table (set) of numbers!
We will also agree do not rearrange number, unless otherwise stated in the explanation. Each number has its own location, and you cannot shuffle them!
The matrix in question has two rows: 
and three columns: 
STANDARD: when talking about the dimensions of the matrix, then first indicate the number of rows, and only then - the number of columns. We have just broken down the two-by-three matrix.
If the number of rows and columns of a matrix is the same, then the matrix is called square, for example: 
is 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 "y": . Essentially, the coordinates of a point are written into a one-by-two matrix. By the way, here is an example for you why the order of numbers matters: and are two completely different points of the plane.
Now let's move on to the study. matrix operations:
1) Action one. Removing a minus from a matrix (Introducing a minus into a matrix).
Back to our matrix 
. As you probably noticed, there are too many negative numbers in this matrix. This is very inconvenient in terms of performing various actions with the matrix, it is inconvenient to write so many minuses, and it just looks ugly in the design.
Let's move the minus outside the matrix by changing the sign of EACH element of the matrix:
At zero, as you understand, the sign does not change, zero - it is also zero in Africa.
Reverse example: 
. Looks ugly.
We introduce a minus into the matrix by changing the sign of EACH element of the matrix:
Well, it's much prettier. And, most importantly, it will be EASIER to perform any actions with the matrix. Because there is such a mathematical folk sign: the more minuses - the more confusion and errors.
2) Action two. Multiplying a Matrix by a Number.
Example:
It's simple, in order to multiply a matrix by a number, you need each multiply the matrix element by the given number. In this case, three.
Another useful example:
– multiplication of a matrix by a fraction
Let's first look at what to do NO NEED:
It is NOT NECESSARY to enter a fraction into the matrix, firstly, it only makes further actions with the matrix difficult, 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 the article Mathematics for dummies or where to start, we remember that decimal fractions with a comma in higher mathematics are trying in every possible way to avoid.
The only thing desirable to do in this example is to insert a minus into the matrix:
But if ALL matrix elements were divided by 7 without a trace, then it would be possible (and necessary!) to divide.
Example:
In this case, you can NEED multiply all elements of the matrix by , since all numbers in the matrix are divisible by 2 without a trace.
Note: in the theory of higher mathematics there is no school concept of "division". Instead of the phrase "this is divided by this", you can always say "this is multiplied by a fraction." That is, division is a special case of multiplication.
3) Action three. Matrix transposition.
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 in a column:
is the transposed matrix.
The transposed matrix is usually denoted by a superscript or a stroke on the top right.
Step by step example:
Transpose Matrix 
First, we rewrite the first row into the first column:
Then we rewrite the second row into the second column: 
And finally, we rewrite the third row into the third column:
Ready. Roughly speaking, to transpose means to turn the matrix on its side.
4) Action four. Sum (difference) of matrices.
The sum of matrices is a simple operation.
NOT ALL MATRIXES CAN BE FOLDED. To perform 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 to a two-by-two matrix and no other! 
Example:
Add matrices 
and 
To add matrices, you need 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 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 will add a minus to the matrix:
Note: in the theory of higher mathematics there is no school concept of "subtraction". Instead of the phrase “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?
For a matrix to be multiplied by a matrix, 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?
So, you can multiply the data of the matrix.
But if the matrices are rearranged, then, in this case, multiplication is no longer possible!
Therefore, multiplication is impossible:
It is not uncommon for tasks with a trick, 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 in both ways.
For example, for matrices, and both multiplication and multiplication are possible
1st year, higher mathematics, study matrices and basic actions on them. Here we systematize the main operations that can be performed with matrices. How to get started 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!
Matrix definition
Matrix is a rectangular table of elements. Well, if in simple terms - a table of numbers.
Matrices are usually denoted by uppercase Latin letters. For example, matrix A , matrix B and so on. 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 the n , where m is the number of lines, and n is the number of columns.
Elements for which i=j (a11, a22, .. ) form the main diagonal of the matrix, and are called diagonal.
What can be done 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 corresponding elements . Let's take 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 by this number each of its elements. 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 with each other. 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. Moreover, each element of the resulting matrix 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 the matrices:
Matrix transpose operation
Matrix transposition is an operation where the corresponding rows and columns are swapped. For example, we transpose the matrix A from the first example:
Matrix determinant
The determinant, oh the determinant, is one of the basic concepts of linear algebra. Once upon a time, people came up with linear equations, and after them they had to invent 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 the 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 it can be done.
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 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 triangles with a face parallel to the secondary diagonal are subtracted.
Fortunately, it is rarely necessary to calculate the determinants of large matrices in practice.
Here we have considered the basic operations on matrices. Of course, in real life you can never even come across a hint of a matrix system of equations, or vice versa, you may encounter much more complex cases when you really have to rack your brains. It is for such cases that there is a professional student service. Ask for help, get a high-quality and detailed solution, enjoy academic success and free time.
This topic will cover operations such as addition and subtraction of matrices, multiplication of a matrix by a number, multiplication of a matrix by a matrix, matrix transposition. All symbols used on this page are taken from the previous topic.
Addition and subtraction of matrices.
The sum $A+B$ of the 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) $.
A similar definition is introduced for the difference of matrices:
The difference $A-B$ of the 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 for the entry $i=\overline(1,m)$: show\hide
The entry "$i=\overline(1,m)$" means that the parameter $i$ changes from 1 to m. For example, the entry $i=\overline(1,5)$ says that the $i$ parameter takes the values 1, 2, 3, 4, 5.
It is worth noting that addition and subtraction operations are defined only for matrices of the same size. In general, the addition and subtraction of matrices are operations that are intuitively clear, because they mean, in fact, just the summation or subtraction of the corresponding elements.
Example #1
Three matrices are given:
$$ A=\left(\begin(array) (ccc) -1 & -2 & 1 \\ 5 & 9 & -8 \end(array) \right)\;\; B=\left(\begin(array) (ccc) 10 & -25 & 98 \\ 3 & 0 & -14 \end(array) \right); \;\; F=\left(\begin(array) (cc) 1 & 0 \\ -5 & 4 \end(array) \right). $$
Is it possible to find the matrix $A+F$? Find matrices $C$ and $D$ if $C=A+B$ and $D=A-B$.
Matrix $A$ contains 2 rows and 3 columns (in other words, the size of matrix $A$ is $2\times 3$), and matrix $F$ contains 2 rows and 2 columns. The dimensions of the matrix $A$ and $F$ do not match, so we cannot add them, i.e. the operation $A+F$ for these matrices is not defined.
The sizes of the matrices $A$ and $B$ are the same, i.e. 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 $D=A-B$:
$$ D=A-B=\left(\begin(array) (ccc) -1 & -2 & 1 \\ 5 & 9 & -8 \end(array) \right)- \left(\begin(array) ( ccc) 10 & -25 & 98 \\ 3 & 0 & -14 \end(array) \right)=\\= \left(\begin(array) (ccc) -1-10 & -2-(-25 ) & 1-98 \\ 5-3 & 9-0 & -8-(-14) \end(array) \right)= \left(\begin(array) (ccc) -11 & 23 & -97 \ \ 2 & 9 & 6 \end(array) \right) $$
Answer: $C=\left(\begin(array) (ccc) 9 & -27 & 99 \\ 8 & 9 & -22 \end(array) \right)$, $D=\left(\begin(array) (ccc) -11 & 23 & -97 \\ 2 & 9 & 6 \end(array) \right)$.
Multiplying a matrix by a number.
The product of the matrix $A_(m\times n)=(a_(ij))$ and 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, to multiply a matrix by some number means to multiply each element of the given matrix by that number.
Example #2
Given a matrix: $ A=\left(\begin(array) (ccc) -1 & -2 & 7 \\ 4 & 9 & 0 \end(array) \right)$. Find 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 notation $-A$ is 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: $3\cdot A=\left(\begin(array) (ccc) -3 & -6 & 21 \\ 12& 27 & 0 \end(array) \right);\; -5\cdot A=\left(\begin(array) (ccc) 5 & 10 & -35 \\ -20 & -45 & 0 \end(array) \right);\; -A=\left(\begin(array) (ccc) 1 & 2 & -7 \\ -4 & -9 & 0 \end(array) \right)$.
The product of two matrices.
The definition of this operation is cumbersome and, at first glance, incomprehensible. Therefore, I will first indicate a general definition, and then we will analyze in detail what it means and how to work with it.
The product of the matrix $A_(m\times n)=(a_(ij))$ and the matrix $B_(n\times k)=(b_(ij))$ is 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 $A$ and the elements of the j-th column of the matrix $B$: $$c_(ij)=\sum\limits_ (p=1)^(n)a_(ip)b_(pj), \;\; i=\overline(1,m), j=\overline(1,n).$$
Step by step, we will analyze the multiplication of matrices using an example. However, you should immediately pay attention that not all matrices can be multiplied. If we want to multiply matrix $A$ by matrix $B$, then first we need to make sure that the number of columns of matrix $A$ is equal to the number of rows of matrix $B$ (such matrices are often called agreed). For example, matrix $A_(5\times 4)$ (matrix contains 5 rows and 4 columns) cannot be multiplied by matrix $F_(9\times 8)$ (9 rows and 8 columns), since the number of columns of matrix $A $ is not equal to the number of rows of matrix $F$, i.e. $4\neq 9$. But it is possible to multiply the matrix $A_(5\times 4)$ by the matrix $B_(4\times 9)$, since the number of columns of the matrix $A$ is equal to the number of rows of the matrix $B$. In this case, the result of multiplying the matrices $A_(5\times 4)$ and $B_(4\times 9)$ is the matrix $C_(5\times 9)$, containing 5 rows and 9 columns:
Example #3
Given matrices: $ 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, we immediately determine the size of the matrix $C$. Since matrix $A$ has size $3\times 4$ and matrix $B$ has size $4\times 2$, the size of matrix $C$ 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 the $c_(11)$ element. 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 $B$:
Similarly 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 the matrix $C$ are found. We pass to the second line, which begins with the element $c_(21)$. 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 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 elements of the matrix $C$ are found, it remains only to write down that $C=\left(\begin(array) (cc) 0 & 37 \\ -23 & 91 \\ 8 & 216 \end(array) \right)$ . Or, to write it 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 location of each element of the result matrix. For matrices, the size of which is small, you can do the following:
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 permutational(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 one or another matrix: on the right or on the left. For example, the phrase "multiply both sides of the equality $3E-F=Y$ by the matrix $A$ on the right" means that you want to get the following equality: $(3E-F)\cdot A=Y\cdot A$.
Transposed with respect to the matrix $A_(m\times n)=(a_(ij))$ is the matrix $A_(n\times m)^(T)=(a_(ij)^(T))$, for elements where $a_(ij)^(T)=a_(ji)$.
Simply put, in order to get the transposed matrix $A^T$, you need to replace the columns in the original matrix $A$ with the corresponding rows according to this principle: there was the first row - the first column will become; there was a second row - the second column will become; there was a third row - 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 had size $3\times 5$, then the transposed matrix has size $5\times 3$.
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.
- $A+B=B+A$ (commutativity of addition)
- $A+(B+C)=(A+B)+C$ (addition associativity)
- $(\alpha+\beta)\cdot A=\alpha A+\beta A$ (distributivity of multiplication by a matrix with respect to addition of numbers)
- $\alpha\cdot(A+B)=\alpha A+\alpha B$ (distributivity of multiplication by a number with respect to matrix addition)
- $A(BC)=(AB)C$
- $(\alpha\beta)A=\alpha(\beta A)$
- $A\cdot (B+C)=AB+AC$, $(B+C)\cdot A=BA+CA$.
- $A\cdot E=A$, $E\cdot A=A$, where $E$ is the identity matrix of the corresponding order.
- $A\cdot O=O$, $O\cdot A=O$, where $O$ is a zero matrix of the appropriate size.
- $\left(A^T \right)^T=A$
- $(A+B)^T=A^T+B^T$
- $(AB)^T=B^T\cdot A^T$
- $\left(\alpha A \right)^T=\alpha A^T$
In the next part, the operation of raising a matrix to a non-negative integer power will be considered, and examples will be solved in which several operations on matrices will be required.
Lecture №1
MATRIX
Definition and types of matrices
Definition 1.1.Matrix size t P is 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. Matrix elements can be functions. To designate the elements of the matrix, lowercase letters of the Latin alphabet with double indexing are used: aij, where is the first index i(read - and) - line number, second index j(read - live) – column number.
Definition 1.2. The matrix is called square p- order if the number of its rows is equal to the number of columns and is equal to the same number P
For a square matrix, the concepts main and side diagonals.
Definition 1.3.Main Diagonal a square matrix consists of elements that have the same indices, i.e. . These are the elements: a 11,a 22,…
Definition 1.4. diagonal if all elements except 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 P- 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 single row matrix is called row matrix.
Definition 1.9. A matrix with one column is called column matrix.
A = (a 11 a 12 ... a 1n) - matrix-row;
Definition 1.10. Two matrices BUT and AT of the same size are called equal, if all corresponding elements of these matrices are equal, i.e. aij = bij for any i= 1, 2, ..., t; j = 1, 2,…, n.
Matrix operations
On matrices, as well as on numbers, a number of operations can be performed. The main operations on matrices are addition (subtraction) of matrices, multiplication of a matrix by a number, and multiplication of matrices. 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 BUT to the number λ .
Example 1.1. Find the product of a matrix A= 
to number 5.
Solution. .◄ 5A= 
Rule for multiplying a matrix by a number: to multiply a matrix by a number, you need to multiply all the elements of the matrix by that number.
Consequence.
1. The common factor of all elements of the matrix can be taken out of the sign of the matrix.
2. Matrix product BUT the number 0 has a zero matrix: BUT· 0 = 0 .
Matrix addition
Definition 1.12.The sum of two matrices A and B the same size t n called matrix FROM= BUT+ AT, whose elements are obtained by adding the corresponding elements of the matrix BUT and matrices AT, 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 BUT with a zero matrix is equal to the original matrix: A + O = A.
1.2.3. Matrix subtraction
Difference of two matrices of the same size is determined through the previous operations: A - B \u003d A + (- 1)AT.
Definition 1.13. Matrix –A = (– 1)BUT called opposite matrix BUT.
Consequence. The sum of opposite matrices is equal to the zero matrix : A + (-A) \u003d O.
Matrix multiplication
Definition 1.14.Multiplication of matrix A by matrix B defined when the number of columns of the first matrix is equal to the number of rows of the second matrix. Then matrix product such a matrix is called , each element of which cij is equal to the sum of products of elements i-th row of the matrix BUT on the relevant elements j-th column of the matrix b.
Example 1.4. Calculate the product of matrices A B where
A= 
= 
Example 1.5. Find products of matrices 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, the product of the matrices VA may not exist. Indeed, in Example 1.4 the matrix product 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 exist and both are matrices of the same size (this is possible only when multiplying square matrices of the same order), then the commutative (displacement) law of multiplication still does not hold, those. A B In A, as in example 1.5;
3) however, if we multiply the square matrix BUT to the identity matrix E the same order, then AE = EA = A.
Thus, the identity matrix plays the same role in matrix multiplication as the number 1 plays in number multiplication;
4) the product of two non-zero matrices can be equal to the zero matrix, i.e. from the fact that A B= 0, it does not follow that A = 0 or B= 0.
