A matrix that has all elements is called diagonal. Mathematics for dummies. Matrices and basic operations on them

Points in space, product Rv gives another vector that determines the position of the point after rotation. If v is a row vector, the same transformation can be obtained using vR T, where R T - transposed to R matrix.

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Matrix: definition and basic concepts

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Sum and difference of matrices, multiplication of a matrix by a number

Transposed matrix / Transposed matrix

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Main diagonal

Elements a ii (i = 1, ..., n) form the main diagonal of a square matrix. These elements lie on an imaginary straight line running from the upper left corner to the lower right corner of the matrix. For example, the main diagonal of the 4x4 matrix in the figure contains the elements a 11 = 9, a 22 = 11, a 33 = 4, a 44 = 10.

The diagonal of a square matrix passing through the lower left and upper right corners is called side.

Special types

Name	Example with n = 3
Diagonal matrix	[ a 11 0 0 0 a 22 0 0 0 a 33 ] (\displaystyle (\begin(bmatrix)a_(11)&0&0\\0&a_(22)&0\\0&0&a_(33)\end(bmatrix)))
Lower triangular matrix	[ a 11 0 0 a 21 a 22 0 a 31 a 32 a 33 ] (\displaystyle (\begin(bmatrix)a_(11)&0&0\\a_(21)&a_(22)&0\\a_(31)&a_( 32)&a_(33)\end(bmatrix)))
Upper triangular matrix	[ a 11 a 12 a 13 0 a 22 a 23 0 0 a 33 ] (\displaystyle (\begin(bmatrix)a_(11)&a_(12)&a_(13)\\0&a_(22)&a_(23)\\ 0&0&a_(33)\end(bmatrix)))

Diagonal and triangular matrices

If all elements outside the main diagonal are zero, A called diagonal. If all elements above (below) the main diagonal are zero, A called the lower (upper) triangular matrix.

Identity matrix

Q(x) = x T Ax

takes only positive values (respectively, negative values or both). If a quadratic form takes only non-negative (respectively, only non-positive) values, the symmetric matrix is called positively semidefinite (respectively, negative semidefinite). A matrix will be indeterminate if it is neither positive nor negative semidefinite.

A symmetric matrix is positive definite if and only if all of its eigenvalues are positive. The table on the right shows two possible cases for 2x2 matrices.

If we use two different vectors, we obtain a bilinear form associated with A:

B A (x, y) = x T Ay.

Orthogonal matrix

Orthogonal matrix is a square matrix with real elements whose columns and rows are orthogonal unit vectors (i.e., orthonormal). You can also define an orthogonal matrix as a matrix whose inverse is equal to its transpose:

A T = A − 1 , (\displaystyle A^(\mathrm (T) )=A^(-1),)

where does it come from

A T A = A A T = E (\displaystyle A^(T)A=AA^(T)=E),

Orthogonal matrix A always reversible ( A −1 = A T), unitary ( A −1 = A*), and normal ( A*A = A.A.*). The determinant of any orthonormal matrix is either +1 or −1. As a linear mapping, any orthonormal matrix with determinant +1 is a simple rotation, while any orthonormal matrix with determinant −1 is either a simple reflection or a composition of reflection and rotation.

Operations

Track

Determinant det( A) or | A| square matrix A is a number that determines some properties of the matrix. A matrix is invertible if and only if its determinant is nonzero.

In this topic we will consider the concept of a matrix, as well as types of matrices. Since there are a lot of terms in this topic, I will add summary to make it easier to navigate the material.

Definition of a matrix and its element. Notation.

Matrix is a table of $m$ rows and $n$ columns. The elements of a matrix can be objects of a completely different nature: numbers, variables or, for example, other matrices. For example, the matrix $\left(\begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right)$ contains 3 rows and 2 columns; its elements are integers. The matrix $\left(\begin(array) (cccc) a & a^9+2 & 9 & \sin x \\ -9 & 3t^2-4 & u-t & 8\end(array) \right)$ contains 2 rows and 4 columns.

Different ways to write matrices: show\hide

The matrix can be written not only in round, but also in square or double straight brackets. That is, the entries below mean the same matrix:

$$ \left(\begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right);\;\; \left[ \begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right]; \;\; \left \Vert \begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right \Vert $$

The product $m\times n$ is called matrix size. For example, if a matrix contains 5 rows and 3 columns, then we speak of a matrix of size $5\times 3$. The matrix $\left(\begin(array)(cc) 5 & 3\\0 & -87\\8 & 0\end(array)\right)$ has size $3 \times 2$.

Typically, matrices are denoted by capital letters of the Latin alphabet: $A$, $B$, $C$ and so on. For example, $B=\left(\begin(array) (ccc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right)$. Line numbering goes from top to bottom; columns - from left to right. For example, the first row of matrix $B$ contains elements 5 and 3, and the second column contains elements 3, -87, 0.

Elements of matrices are usually denoted in small letters. For example, the elements of the matrix $A$ are denoted by $a_(ij)$. The double index $ij$ contains information about the position of the element in the matrix. The number $i$ is the row number, and the number $j$ is the column number, at the intersection of which is the element $a_(ij)$. For example, at the intersection of the second row and the fifth column of the matrix $A=\left(\begin(array) (cccccc) 51 & 37 & -9 & 0 & 9 & 97 \\ 1 & 2 & 3 & 41 & 59 & 6 \ \ -17 & -15 & -13 & -11 & -8 & -5 \\ 52 & 31 & -4 & -1 & 17 & 90 \end(array) \right)$ element $a_(25)= $59:

In the same way, at the intersection of the first row and the first column we have the element $a_(11)=51$; at the intersection of the third row and the second column - the element $a_(32)=-15$ and so on. Note that the entry $a_(32)$ reads “a three two”, but not “a thirty two”.

To abbreviate the matrix $A$, the size of which is $m\times n$, the notation $A_(m\times n)$ is used. You can write it in a little more detail:

$$ A_(m\times n)=(a_(ij)) $$

where the notation $(a_(ij))$ denotes the elements of the matrix $A$. In its fully expanded form, the matrix $A_(m\times n)=(a_(ij))$ can be written as follows:

$$ A_(m\times n)=\left(\begin(array)(cccc) a_(11) & a_(12) & \ldots & a_(1n) \\ a_(21) & a_(22) & \ldots & a_(2n) \\ \ldots & \ldots & \ldots & \ldots \\ a_(m1) & a_(m2) & \ldots & a_(mn) \end(array) \right) $$

Let's introduce another term - equal matrices.

Two matrices of the same size $A_(m\times n)=(a_(ij))$ and $B_(m\times n)=(b_(ij))$ are called equal, if their corresponding elements are equal, i.e. $a_(ij)=b_(ij)$ for all $i=\overline(1,m)$ and $j=\overline(1,n)$.

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

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

So, for matrices to be equal, two conditions must be met: coincidence of sizes and equality of the corresponding elements. For example, the matrix $A=\left(\begin(array)(cc) 5 & 3\\0 & -87\\8 & 0\end(array)\right)$ is not equal to the matrix $B=\left(\ begin(array)(cc) 8 & -9\\0 & -87 \end(array)\right)$ because matrix $A$ has size $3\times 2$ and matrix $B$ has size $2\times $2. Also, matrix $A$ is not equal to matrix $C=\left(\begin(array)(cc) 5 & 3\\98 & -87\\8 & 0\end(array)\right)$, since $a_( 21)\neq c_(21)$ (i.e. $0\neq 98$). But for the matrix $F=\left(\begin(array)(cc) 5 & 3\\0 & -87\\8 & 0\end(array)\right)$ we can safely write $A=F$ because both the sizes and the corresponding elements of the matrices $A$ and $F$ coincide.

Example No. 1

Determine the size of the matrix $A=\left(\begin(array) (ccc) -1 & -2 & 1 \\ 5 & 9 & -8 \\ -6 & 8 & 23 \\ 11 & -12 & -5 \ \4 & 0 & -10 \\ \end(array) \right)$. Indicate what the elements $a_(12)$, $a_(33)$, $a_(43)$ are equal to.

This matrix contains 5 rows and 3 columns, so its size is $5\times 3$. You can also use the notation $A_(5\times 3)$ for this matrix.

Element $a_(12)$ is at the intersection of the first row and second column, so $a_(12)=-2$. Element $a_(33)$ is at the intersection of the third row and third column, so $a_(33)=23$. Element $a_(43)$ is at the intersection of the fourth row and third column, so $a_(43)=-5$.

Answer: $a_(12)=-2$, $a_(33)=23$, $a_(43)=-5$.

Types of matrices depending on their size. Main and secondary diagonals. Matrix trace.

Let a certain matrix $A_(m\times n)$ be given. If $m=1$ (the matrix consists of one row), then the given matrix is called matrix-row. If $n=1$ (the matrix consists of one column), then such a matrix is called matrix-column. For example, $\left(\begin(array) (ccccc) -1 & -2 & 0 & -9 & 8 \end(array) \right)$ is a row matrix, and $\left(\begin(array) (c) -1 \\ 5 \\ 6 \end(array) \right)$ is a column matrix.

If the matrix $A_(m\times n)$ satisfies the condition $m\neq n$ (i.e., the number of rows is not equal to the number of columns), then it is often said that $A$ is a rectangular matrix. For example, the matrix $\left(\begin(array) (cccc) -1 & -2 & 0 & 9 \\ 5 & 9 & 5 & 1 \end(array) \right)$ has size $2\times 4$, those. contains 2 rows and 4 columns. Since the number of rows is not equal to the number of columns, this matrix is rectangular.

If the matrix $A_(m\times n)$ satisfies the condition $m=n$ (i.e., the number of rows is equal to the number of columns), then $A$ is said to be a square matrix of order $n$. For example, $\left(\begin(array) (cc) -1 & -2 \\ 5 & 9 \end(array) \right)$ is a second-order square matrix; $\left(\begin(array) (ccc) -1 & -2 & 9 \\ 5 & 9 & 8 \\ 1 & 0 & 4 \end(array) \right)$ is a third-order square matrix. IN general view the square matrix $A_(n\times n)$ can be written as follows:

$$ A_(n\times n)=\left(\begin(array)(cccc) a_(11) & a_(12) & \ldots & a_(1n) \\ a_(21) & a_(22) & \ldots & a_(2n) \\ \ldots & \ldots & \ldots & \ldots \\ a_(n1) & a_(n2) & \ldots & a_(nn) \end(array) \right) $$

The elements $a_(11)$, $a_(22)$, $\ldots$, $a_(nn)$ are said to be on main diagonal matrices $A_(n\times n)$. These elements are called main diagonal elements(or just diagonal elements). The elements $a_(1n)$, $a_(2 \; n-1)$, $\ldots$, $a_(n1)$ are on side (minor) diagonal; they are called side diagonal elements. For example, for the matrix $C=\left(\begin(array)(cccc)2&-2&9&1\\5&9&8& 0\\1& 0 & 4 & -7 \\ -4 & -9 & 5 & 6\end(array) \right)$ we have:

The elements $c_(11)=2$, $c_(22)=9$, $c_(33)=4$, $c_(44)=6$ are the main diagonal elements; elements $c_(14)=1$, $c_(23)=8$, $c_(32)=0$, $c_(41)=-4$ are side diagonal elements.

The sum of the main diagonal elements is called followed by the matrix and is denoted by $\Tr A$ (or $\Sp A$):

$$ \Tr A=a_(11)+a_(22)+\ldots+a_(nn) $$

For example, for the matrix $C=\left(\begin(array) (cccc) 2 & -2 & 9 & 1\\5 & 9 & 8 & 0\\1 & 0 & 4 & -7\\-4 & -9 & 5 & 6 \end(array)\right)$ we have:

$$ \Tr C=2+9+4+6=21. $$

The concept of diagonal elements is also used for non-square matrices. For example, for the matrix $B=\left(\begin(array) (ccccc) 2 & -2 & 9 & 1 & 7 \\ 5 & -9 & 8 & 0 & -6 \\ 1 & 0 & 4 & - 7 & -6 \end(array) \right)$ the main diagonal elements will be $b_(11)=2$, $b_(22)=-9$, $b_(33)=4$.

Types of matrices depending on the values of their elements.

If all elements of the matrix $A_(m\times n)$ are equal to zero, then such a matrix is called null and is usually denoted by the letter $O$. For example, $\left(\begin(array) (cc) 0 & 0 \\ 0 & 0 \\ 0 & 0 \end(array) \right)$, $\left(\begin(array) (ccc) 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end(array) \right)$ - zero matrices.

Let the matrix $A_(m\times n)$ have the following form:

Then this matrix is called trapezoidal. It may not contain zero rows, but if they exist, they are located at the bottom of the matrix. In a more general form, a trapezoidal matrix can be written as follows:

Again, trailing null lines are not required. Those. Formally, we can distinguish the following conditions for a trapezoidal matrix:

All elements below the main diagonal are zero.
All elements from $a_(11)$ to $a_(rr)$ lying on the main diagonal are not equal to zero: $a_(11)\neq 0, \; a_(22)\neq 0, \ldots, a_(rr)\neq 0$.
Either all elements of the last $m-r$ rows are zero, or $m=r$ (i.e. there are no zero rows at all).

Examples of trapezoidal matrices:

Let's move on to the next definition. The matrix $A_(m\times n)$ is called stepped, if it satisfies the following conditions:

For example, step matrices will be:

For comparison, the matrix $\left(\begin(array) (cccc) 2 & -2 & 0 & 1\\0 & 0 & 8 & 7\\0 & 0 & 4 & -7\\0 & 0 & 0 & 0 \end(array)\right)$ is not echelon because the third row has the same zero part as the second row. That is, the principle “the lower the line, the larger the zero part” is violated. I will add that a trapezoidal matrix is a special case of a stepped matrix.

Let's move on to the next definition. If all elements of a square matrix located under the main diagonal are equal to zero, then such a matrix is called upper triangular matrix. For example, $\left(\begin(array) (cccc) 2 & -2 & 9 & 1 \\ 0 & 9 & 8 & 0 \\ 0 & 0 & 4 & -7 \\ 0 & 0 & 0 & 6 \end(array) \right)$ is an upper triangular matrix. Note that the definition of an upper triangular matrix does not say anything about the values of the elements located above the main diagonal or on the main diagonal. They can be zero or not - it doesn't matter. For example, $\left(\begin(array) (ccc) 0 & 0 & 9 \\ 0 & 0 & 0\\ 0 & 0 & 0 \end(array) \right)$ is also an upper triangular matrix.

If all elements of a square matrix located above the main diagonal are equal to zero, then such a matrix is called lower triangular matrix. For example, $\left(\begin(array) (cccc) 3 & 0 & 0 & 0 \\ -5 & 1 & 0 & 0 \\ 8 & 2 & 1 & 0 \\ 5 & 4 & 0 & 6 \ end(array) \right)$ - lower triangular matrix. Note that the definition of a lower triangular matrix does not say anything about the values of the elements located under or on the main diagonal. They may be zero or not - it doesn't matter. For example, $\left(\begin(array) (ccc) -5 & 0 & 0 \\ 0 & 0 & 0\\ 0 & 0 & 9 \end(array) \right)$ and $\left(\begin (array) (ccc) 0 & 0 & 0 \\ 0 & 0 & 0\\ 0 & 0 & 0 \end(array) \right)$ are also lower triangular matrices.

The square matrix is called diagonal, if all elements of this matrix that do not lie on the main diagonal are equal to zero. Example: $\left(\begin(array) (cccc) 3 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 \ end(array)\right)$. The elements on the main diagonal can be anything (equal to zero or not) - it doesn't matter.

The diagonal matrix is called single, if all elements of this matrix located on the main diagonal are equal to 1. For example, $\left(\begin(array) (cccc) 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end(array)\right)$ - fourth-order identity matrix; $\left(\begin(array) (cc) 1 & 0 \\ 0 & 1 \end(array)\right)$ is the second-order identity matrix.

Given Toolkit 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 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 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 on the 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 by 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 this case - a 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 the matrix, and secondly, it makes it difficult for the teacher to check the solution (especially if – 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 decimals in higher mathematics they try to avoid them 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 a special case of 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 can you solve this example more easily, 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.” a 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

Operations on matrices and their properties.

The concept of a determinant of the second and third orders.Properties of determinants and their calculation.

3. general description tasks.

4. Completing tasks.

5. Preparation of a report on laboratory work.

Glossary

Learn the definitions of the following terms:

Dimension A matrix is a collection of two numbers, consisting of the number of its rows m and the number of columns n.

If m=n, then the matrix is called square matrix of order n.

Operations on matrices: transposing a matrix, multiplying (dividing) a matrix by a number, adding and subtracting, multiplying a matrix by a matrix.

The transition from a matrix A to a matrix A m, the rows of which are the columns, and the columns are the rows of the matrix A, is called transposition matrices A.

Example: A = , A t = .

To multiply matrix by number, you need to multiply each element of the matrix by this number.

Example: 2A= 2· = .

Sum (difference) matrices A and B of the same dimension are called matrix C=A B, the elements of which are equal with ij = a ij b ij for all i And j.

Example: A = ; B = . A+B= = .

The work matrix A m n into matrix B n k is called matrix C m k , each element of which is c ij equal to the sum products of the elements of the i-th row of matrix A by the corresponding element of the j-th column of matrix B:

c ij = a i1 · b 1j + a i2 ·b 2j +…+ a in ·b nj .

To be able to multiply a matrix by a matrix, they must be agreed upon for multiplication, namely number of columns in the first matrix should be equal to number of lines in the second matrix.

Example: A= and B=.

А·В—impossible, because they are not consistent.

VA= . = = .

Properties of the matrix multiplication operation.

1. If matrix A has the dimension m n, and matrix B is the dimension n k, then the product A·B exists.

The product BA can exist only when m=k.

2. Matrix multiplication is not commutative, i.e. A·B is not always equal to BA·A even if both products are defined. However, if the relation А·В=В·А is satisfied, then the matrices A and B are called permutable.

Example. Calculate.

Minor element is the determinant of the order matrix, obtained by deleting the th row of the th column.

Algebraic complement element is called .

Laplace expansion theorem:

The determinant of a square matrix is equal to the sum of the products of the elements of any row (column) by their algebraic complements.

Example. Calculate.

Solution. .

Properties of nth order determinants:

1) The value of the determinant will not change if the rows and columns are swapped.

2) If the determinant contains a row (column) of only zeros, then it is equal to zero.

3) When rearranging two rows (columns), the determinant changes sign.

4) A determinant that has two identical rows (columns) is equal to zero.

5) The common factor of the elements of any row (column) can be taken out of the determinant sign.

6) If each element of a certain row (column) is the sum of two terms, then the determinant is equal to the sum of two determinants, in each of which all rows (columns), except the one mentioned, are the same as in this determinant, and in the mentioned row ( Column) of the first determinant contains the first terms, the second - the second.

7) If two rows (columns) in the determinant are proportional, then it is equal to zero.

8) The determinant will not change if the corresponding elements of another row (column) are added to the elements of a certain row (column), multiplied by the same number.

9) The determinants of triangular and diagonal matrices are equal to the product of the elements of the main diagonal.

The method of accumulating zeros for calculating determinants is based on the properties of determinants.

Example. Calculate.

Solution. Subtract the double third from the first row, then use the expansion theorem in the first column.

~ .

Control questions(OK-1, OK-2, OK-11, PK-1) :

1. What is called a second-order determinant?

2. What are the main properties of determinants?

3. What is the minor of an element?

4. What is called the algebraic complement of an element of a determinant?

5. How to expand the third-order determinant into elements of a row (column)?

6. What is the sum of the products of the elements of a row (or column), the determinant of the algebraic complements of the corresponding elements of another row (or column)?

7. What is the rule of triangles?

8. How are determinants of higher orders calculated using the order reduction method?

10. Which matrix is called square? Null? What is a row matrix, column matrix?

11. Which matrices are called equal?

12. Give definitions of the operations of addition, multiplication of matrices, multiplication of a matrix by a number

13. What conditions must the sizes of matrices satisfy during addition and multiplication?

14. What are the properties of algebraic operations: commutativity, associativity, distributivity? Which of them are fulfilled for matrices during addition and multiplication, and which are not?

15. What is inverse matrix? For what matrices is it defined?

16. Formulate a theorem on the existence and uniqueness of the inverse matrix.

17. Formulate a lemma on the transposition of a product of matrices.

General practical tasks(OK-1, OK-2, OK-11, PK-1) :

No. 1. Find the sum and difference of matrices A and B :

No. 2. Follow these steps :

c) Z= -11A+7B-4C+D

No. 3. Follow these steps :

No. 4. Using four methods of calculating the determinant of a square matrix, find the determinants of the following matrices :

No. 5. Find determinants of the nth order, based on the elements of the column (row) :

A) b)

No. 6. Find the determinant of a matrix using the properties of determinants:

A) b)

Matrices in mathematics are one of the most important objects of practical importance. Often an excursion into the theory of matrices begins with the words: “A matrix is a rectangular table...”. We will start this excursion from a slightly different direction.

Phone books of any size and with any amount of subscriber data are nothing more than matrices. Such matrices look approximately like this:

It is clear that we all use such matrices almost every day. These matrices come with a different number of rows (they vary like a directory issued by a telephone company, which can have thousands, hundreds of thousands and even millions of lines, and a new notebook you just started, which has less than ten lines) and columns (a directory of officials of some kind). some organization in which there may be columns such as position and office number and your same address book, where there may not be any data except the name, and thus there are only two columns in it - name and telephone number).

All sorts of matrices can be added and multiplied, as well as other operations can be performed on them, but there is no need to add and multiply telephone directories, there is no benefit from this, and besides, you can use your mind.

But many matrices can and should be added and multiplied and thus solve various pressing problems. Below are examples of such matrices.

Matrices in which the columns are the production of units of a particular type of product, and the rows are the years in which the production of this product is recorded:

You can add matrices of this type, which take into account the output of similar products by different enterprises, in order to obtain summary data for the industry.

Or matrices consisting, for example, of one column, in which the rows are the average cost of a particular type of product:

The last two types of matrices can be multiplied, and the result is a row matrix containing the cost of all types of products by year.

Matrices, basic definitions

A rectangular table consisting of numbers arranged in m lines and n columns is called mn-matrix (or simply matrix ) and is written like this:

(1)

In matrix (1) the numbers are called its elements (as in the determinant, the first index means the number of the row, the second – the column at the intersection of which the element stands; i = 1, 2, ..., m; j = 1, 2, n).

The matrix is called rectangular , If .

If m = n, then the matrix is called square , and the number n is its in order .

Determinant of a square matrix A is a determinant whose elements are the elements of a matrix A. It is indicated by the symbol | A|.

The square matrix is called not special (or non-degenerate , non-singular ), if its determinant is not zero, and special (or degenerate , singular ) if its determinant is zero.

The matrices are called equal , if they have the same number of rows and columns and all corresponding elements match.

The matrix is called null , if all its elements are equal to zero. We will denote the zero matrix by the symbol 0 or .

For example,

Matrix-row (or lowercase ) is called 1 n-matrix, and matrix-column (or columnar ) – m 1-matrix.

Matrix A", which is obtained from the matrix A swapping rows and columns in it is called transposed relative to the matrix A. Thus, for matrix (1) the transposed matrix is

Matrix transition operation A" transposed with respect to the matrix A, is called matrix transposition A. For mn-matrix transposed is nm-matrix.

The matrix transposed with respect to the matrix is A, that is

(A")" = A .

Example 1. Find matrix A" , transposed with respect to the matrix

and find out whether the determinants of the original and transposed matrices are equal.

Main diagonal A square matrix is an imaginary line connecting its elements, for which both indices are the same. These elements are called diagonal .

A square matrix in which all elements off the main diagonal are equal to zero is called diagonal . Not all diagonal elements of a diagonal matrix are necessarily nonzero. Some of them may be equal to zero.

A square matrix in which the elements on the main diagonal are equal to the same number, non-zero, and all others are equal to zero, is called scalar matrix .

Identity matrix is called a diagonal matrix in which all diagonal elements are equal to one. For example, the third-order identity matrix is the matrix

Example 2. Given matrices:

Solution. Let us calculate the determinants of these matrices. Using the triangle rule, we find

Matrix determinant B let's calculate using the formula

We easily get that

Therefore, the matrices A and are non-singular (non-degenerate, non-singular), and the matrix B– special (degenerate, singular).

The determinant of the identity matrix of any order is obviously equal to one.

Solve the matrix problem yourself, and then look at the solution

Example 3. Given matrices

Determine which of them are non-singular (non-degenerate, non-singular).

Application of matrices in mathematical and economic modeling

Structured data about a particular object is simply and conveniently recorded in the form of matrices. Matrix models are created not only to store this structured data, but also to solve various problems with this data using linear algebra.

Thus, a well-known matrix model of the economy is the input-output model, introduced by the American economist of Russian origin Vasily Leontiev. This model is based on the assumption that the entire production sector of the economy is divided into n clean industries. Each industry produces only one type of product, and different industries produce different products. Due to this division of labor between industries, there are inter-industry connections, the meaning of which is that part of the production of each industry is transferred to other industries as a production resource.

Product volume i-th industry (measured by a specific unit of measurement), which was produced during the reporting period, is denoted by and is called full output i-th industry. Issues can be conveniently placed in n-component row of the matrix.

Number of units i-industry that needs to be spent j-industry for the production of a unit of its output is designated and called the direct cost coefficient.