Multiplication of two matrices example. Multiplying a square matrix by a column matrix

This topic will cover operations such as adding and subtracting matrices, multiplying a matrix by a number, multiplying a matrix by a matrix, and transposing a matrix. All symbols used on this page are taken from the previous topic.

Addition and subtraction of matrices.

The sum of $A+B$ of matrices $A_(m\times n)=(a_(ij))$ and $B_(m\times n)=(b_(ij))$ is called 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 between the $A-B$ 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 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.

It is worth noting that addition and subtraction operations are defined only for matrices of the same size. In general, addition and subtraction of matrices are operations that are clear intuitively, because they essentially mean just the summation or subtraction of the corresponding elements.

Example No. 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 sizes of the matrices $A$ and $F$ do not coincide, so we cannot add them, i.e. the $A+F$ operation is not defined for these matrices.

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.

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

Let's 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 $A_(m\times n)=(a_(ij))$ 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 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 a shorthand notation for $-1\cdot A$. That is, to find $-A$ you need to multiply all elements of the matrix $A$ by (-1). Essentially, 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, unclear. Therefore, I will first point out general definition, and then we’ll look in detail at what it means and how to work with it.

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

Let's look at matrix multiplication step by step using an example. However, you should immediately note that not all matrices can be multiplied. If we want to multiply matrix $A$ by matrix $B$, then we first 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 upon). 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 of the matrix $F$, 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 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 $B_(4\times 9)$ will be the matrix $C_(5\times 9)$, containing 5 rows and 9 columns:

Example No. 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 $C=A\cdot B$.

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

So, as a result of the product of matrices $A$ and $B$, we should obtain matrix $C$, consisting of three lines 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 designation of elements raises questions, then you can look at the previous topic: “Matrixes. Types of matrices. Basic terms”, at the beginning of which the designation of matrix elements is explained. Our goal: to find the values of all elements of the matrix $C$.

Let's start with the element $c_(11)$. To obtain 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 will have to multiply the elements of the first row of matrix $A$ and the second column of 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 matrix $C$ have been found. Let's move on to the second line, which begins with the element $c_(21)$. To find it, you will have to multiply the elements of the second row of matrix $A$ and the first column of matrix $B$:

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

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

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

To find $c_(31)$, multiply the elements of the third row of matrix $A$ by the elements of the first column of 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 elements of the matrix $C$ have been found, all that remains is 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 location of each element of the result matrix. For matrices whose size is small, you can do this:

It is also worth noting that matrix multiplication is non-commutative. This means that in the general case $A\cdot B\neq B\cdot A$. Only for some types of matrices, which are called permutable(or commuting), the equality $A\cdot B=B\cdot A$ is true. It is precisely based on the non-commutativity of multiplication that we need to indicate exactly how we multiply the expression by a particular 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 which $a_(ij)^(T)=a_(ji)$.

Simply put, in order to obtain a 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 a first row - there will be a first column; there was a second line - there will be a second column; 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 a size of $3\times 5$, then the transposed matrix has a size of $5\times 3$.

Some properties of operations on matrices.

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

$A+B=B+A$ (commutativity of addition)
$A+(B+C)=(A+B)+C$ (associativity of addition)
$(\alpha+\beta)\cdot A=\alpha A+\beta A$ (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 relative 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 $O$ - zero matrix 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, we will consider the operation of raising a matrix to a non-negative integer power, and also solve examples in which it is necessary to perform several operations on matrices.

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

Matrix Definition

Matrix is a rectangular table of elements. Well, what if in simple language– table of numbers.

Typically matrices are denoted in capitals in Latin letters. For example, matrix A , matrix B and so on. Matrices can be different sizes: rectangular, square, there are also row matrices and column matrices called vectors. The size of the matrix is determined by the number of rows and columns. For example, let's write rectangular matrix size m on n , Where m – number of lines, and n – number of columns.

Items for which i=j (a11, a22, .. ) form the main diagonal of the matrix and are called diagonal.

What can you do with matrices? Add/Subtract, multiply by a number, multiply among themselves, transpose. Now about all these basic operations on matrices in order.

Matrix addition and subtraction operations

Let us immediately warn you that you can only add matrices 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 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 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 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 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 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 secondary diagonal and the product of the elements lying on the triangles with the face of the parallel secondary diagonal are subtracted.

Fortunately, computing determinants of matrices large sizes in practice it is rarely necessary.

Here we looked at basic operations on matrices. Of course, in real life you may never encounter even a hint of 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.

Given methodological manual 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-monitoring and self-testing, 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 non-scientific terms are possible. Lovers of solid theory, please do not engage in criticism, our task is learn to perform operations with matrices.

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

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 advisable to remember the term, it will appear often, it is no coincidence that I used bold font to highlight it.

Designation: matrices are usually denoted in 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'll also agree don't 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 matrix sizes, then at 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: – a three-by-three matrix.

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

In fact, we have known 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 of why the order of numbers matters: and are two completely different points plane.

Now let's move on to studying operations with matrices:

1) Act one. Removing a minus from the matrix (introducing a minus into the matrix).

Let's return to our matrix . As you probably 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 simply looks ugly in design.

Let's move the minus outside the matrix, changing the sign of EACH element of the matrix:

At zero, as you understand, the sign does not change; zero is also zero in Africa.

Reverse example: . It looks ugly.

Let's introduce a minus into the matrix by changing the sign of EACH element of the matrix:

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 sign: the more minuses, the more confusion and errors.

2) Act two. Multiplying a matrix by a number.

Example:

It's simple, in order to multiply a matrix by a number, you need every matrix element multiplied by a given number. IN in this case- for three.

Another useful example:

– multiplying a matrix by a fraction

First let's look at what to do NO NEED:

There is NO NEED to enter a fraction into the matrix, firstly, it only complicates further actions with a matrix, secondly, it makes it difficult for the teacher to check the solution (especially if – final answer of the task).

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

From the article Mathematics for dummies or where to start, we remember that in higher mathematics they try to avoid decimal fractions with commas in every possible way.

The only thing is preferably What to do in this example is to add a minus to the matrix:

But if only 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 TO multiply all matrix elements by , since all matrix numbers are divisible by 2 without a trace.

Note: in the theory of higher school mathematics there is no concept of “division”. Instead of saying “this divided by that,” you can always say “this multiplied by a fraction.” That is, division is special case multiplication.

3) Act three. 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 needs to be written in a column:

– transposed matrix.

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

Step by step example:

Transpose matrix

First we rewrite the first row into the first column:

Then we rewrite the second line into the second column:

And finally, we rewrite the third row into the third column:

Ready. Roughly speaking, transposing means turning the matrix on its side.

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

The sum of matrices is a simple operation.
NOT ALL MATRICES 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 with a two-by-two matrix and no other!

Example:

Add matrices And

In order 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 matrix difference ,

How to decide this example easier so as not to get confused? It is advisable to get rid of unnecessary minuses; to do this, add a minus to the matrix:

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

5) Act five. Matrix multiplication.

What matrices can be multiplied?

In order for a matrix to be multiplied by a matrix, it is necessary so that the number of matrix columns is equal to the number of matrix rows.

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

This means that matrix data can be multiplied.

But if the matrices are rearranged, then, in this case, multiplication is no longer possible!

Therefore, multiplication is not possible:

It is not so rare to encounter tasks with a trick, when the 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

The main applications of matrices are related to the operation multiplication.

Two matrices are given:

A – size mn

B – size n k

Because the length of a row in matrix A coincides with the height of a column in matrix B, you can define a matrix C=AB, which will have dimensions m k. Element matrix C, located in an arbitrary i-th row (i=1,…,m) and an arbitrary j-th column (j=1,…,k), by definition, is equal to the scalar product of two vectors from
:i-th row of matrix A and j-th column of matrix B:

Properties:

How is the operation of multiplying a matrix A by a number λ defined?

The product of A and the number λ is a matrix in which each element is equal to the product of the corresponding element of A and λ. Consequence: Total multiplier all elements of the matrix can be taken out of the matrix sign.

13. Definition of the inverse matrix and its properties.

Definition. If there are square matrices X and A of the same order satisfying the condition:

where E is the identity matrix of the same order as the matrix A, then the matrix X is called reverse to the matrix A and is denoted by A -1.

Properties of inverse matrices

Let us indicate the following properties of inverse matrices:

1) (A -1) -1 = A;

2) (AB) -1 = B -1 A -1

3) (A T) -1 = (A -1) T .

1. If inverse matrix exists, then it is the only one.

2. Not every non-zero square matrix has an inverse.

14. Give the main properties of determinants. Check the validity of the property |AB|=|A|*|B| for matrices

A= and B=

Properties of determinants:

1. If any row of the determinant consists of zeros, then the determinant itself is equal to zero.

2. When rearranging two rows, the determinant is multiplied by -1.

3. The determinant with two identical rows is equal to zero.

4. The common factor of the elements of any row can be taken out of the determinant sign.

5. If the elements of a certain row of determinant A are presented as the sum of two terms, then the determinant itself is equal to the sum of two determinants B and D. In determinant B, the specified line consists of the first terms, in D - of the second terms. The remaining lines of determinants B and D are the same as in A.

6. The value of the determinant will not change if another line is added to one of the lines, multiplied by any number.

7. The sum of the products of the elements of any row by algebraic additions to the corresponding elements of the other row are equal to 0.

8. The determinant of the matrix A is equal to the determinant of the transposed matrix A m, i.e. the determinant does not change when transposed.

15. Define the modulus and argument of a complex number. Write the numbers √3+ in trigonometric formi, -1+ i.

Each complex number z=a+ib can be associated with a vector (a,b)€R 2. The length of this vector equal to √a 2 + b 2 is called modulus of a complex number z and is denoted by |z|. The angle φ between a given vector and the positive direction of the Ox axis is called complex number argument z and is denoted by arg z.

Any complex number z≠0 can be represented as z=|z|(cosφ +isinφ).

This form of writing a complex number is called trigonometric.

√3+i=2(√3/2+1/2i)=2(cosπ/6+isinπ/6);

1+i=2(-√2/2+i√2/2)=2(cosπ/4+isinπ/4).

Each complex number Z = a + ib can be assigned a vector (a; b) belonging to R^2. The length of this vector, equal to KB from a^2 + b^2, is called the modulus of a complex number and is denoted by the modulus Z. The angle between this vector and the positive direction of the Ox axis is called the argument of the complex number (denoted by arg Z).

Matrix addition:

Subtraction and addition of matrices reduces to the corresponding operations on their elements. Matrix addition operation entered only for matrices the same size, i.e. for matrices, in which the number of rows and columns is respectively equal. Sum of matrices A and B are called matrix C, whose elements are equal to the sum of the corresponding elements. C = A + B c ij = a ij + b ij Defined similarly matrix difference.

Multiplying a matrix by a number:

Matrix multiplication (division) operation of any size by an arbitrary number is reduced to multiplying (dividing) each element matrices for this number. Matrix product And the number k is called matrix B, such that

b ij = k × a ij . B = k × A b ij = k × a ij . Matrix- A = (-1) × A is called the opposite matrix A.

Properties of adding matrices and multiplying a matrix by a number:

Matrix addition operations And matrix multiplication per number have the following properties: 1. A + B = B + A; 2. A + (B + C) = (A + B) + C; 3. A + 0 = A; 4. A - A = 0; 5. 1 × A = A; 6. α × (A + B) = αA + αB; 7. (α + β) × A = αA + βA; 8. α × (βA) = (αβ) × A; , where A, B and C are matrices, α and β are numbers.

Matrix multiplication (Matrix product):

Operation of multiplying two matrices is entered only for the case when the number of columns of the first matrices equal to the number of lines of the second matrices. Matrix product And m×n on matrix In n×p, called matrix With m×p such that with ik = a i1 × b 1k + a i2 × b 2k + ... + a in × b nk , i.e., the sum of the products of the elements of the i-th row is found matrices And to the corresponding elements of the jth column matrices B. If matrices A and B are squares of the same size, then the products AB and BA always exist. It is easy to show that A × E = E × A = A, where A is square matrix, E - unit matrix the same size.

Properties of matrix multiplication:

Matrix multiplication not commutative, i.e. AB ≠ BA even if both products are defined. However, if for any matrices the relationship AB=BA is satisfied, then such matrices are called commutative. The most typical example is a single matrix, which commutes with any other matrix the same size. Only square ones can be permutable matrices of the same order. A × E = E × A = A

Matrix multiplication has the following properties: 1. A × (B × C) = (A × B) × C; 2. A × (B + C) = AB + AC; 3. (A + B) × C = AC + BC; 4. α × (AB) = (αA) × B; 5. A × 0 = 0; 0 × A = 0; 6. (AB) T = B T A T; 7. (ABC) T = C T V T A T; 8. (A + B) T = A T + B T;

2. Determinants of the 2nd and 3rd orders. Properties of determinants.

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.

Determine the number of terms to find determinant of the matrix, in an algebraic sum, you can calculate the factorial: 2! = 1 × 2 = 2 3! = 1 × 2 × 3 = 6

Properties of matrix determinants

Properties of matrix determinants:

Property #1:

Matrix determinant will not change if its rows are replaced with columns, each row with a column with the same number, and vice versa (Transposition). |A| = |A| T

Consequence:

Columns and Rows determinant of the matrix are equal, therefore, the properties inherent in rows are also fulfilled for columns.

Property #2:

When rearranging 2 rows or columns matrix determinant will change the sign to the opposite one, maintaining the absolute value, i.e.:

Property #3:

Matrix determinant having two identical rows is equal to zero.

Property #4:

Common factor of elements of any series determinant of the matrix can be taken as a sign determinant.

Corollaries from properties No. 3 and No. 4:

If all elements of a certain series (row or column) are proportional to the corresponding elements of a parallel series, then such matrix determinant equal to zero.

Property #5:

determinant of the matrix are equal to zero, then matrix determinant equal to zero.

Property #6:

If all elements of a row or column determinant presented as a sum of 2 terms, then determinant matrices can be represented as the sum of 2 determinants according to the formula:

Property #7:

If to any row (or column) determinant add the corresponding elements of another row (or column), multiplied by the same number, then matrix determinant will not change its value.

Example of using properties for calculation determinant of the matrix: