What does the matrix consist of? Algorithm for finding the inverse matrix

Matrices, get acquainted with its basic concepts. The defining elements of a matrix are its diagonals and its side diagonals. Home starts with the element in the first row, first column and continues to the element in the last column, last row (that is, it goes from left to right). The side diagonal begins on the contrary in the first row, but in the last column and continues to the element that has the coordinates of the first column and the last row (goes from right to left).

To move on to the following definitions and algebraic operations with matrices, study the types of matrices. The simplest ones are square, unit, zero and inverse. The number of columns and rows matches. The transposed matrix, let's call it B, is obtained from matrix A by replacing the columns with rows. In unit, all the elements of the main diagonal are ones, and the others are zeros. And in zero, even the elements of the diagonals are zero. The inverse matrix is ​​the one on which the original matrix comes to the identity form.

Also, the matrix can be symmetrical about the main or secondary axes. That is, an element having coordinates a(1;2), where 1 is the row number and 2 is the column number, is equal to a(2;1). A(3;1)=A(1;3) and so on. Matched matrices are those where the number of columns of one is equal to the number of rows of another (such matrices can be multiplied).

The main actions that can be performed with matrices are addition, multiplication and finding the determinant. If the matrices are the same size, that is, they have an equal number of rows and columns, then they can be added. It is necessary to add elements that are in the same places in the matrices, that is, add a (m; n) with c in (m; n), where m and n are the corresponding coordinates of the column and row. When adding matrices, the main rule of ordinary arithmetic addition applies - when the places of the terms are changed, the sum does not change. Thus, if instead of a simple element a there is an expression a + b, then it can be added to an element c of another commensurate matrix according to the rules a + (b + c) = (a + b) + c.

You can multiply the matched matrices given above. This produces a matrix where each element is the sum of the pairwise multiplied elements of a row of matrix A and a column of matrix B. When multiplying, the order of actions is very important. m*n is not equal to n*m.

Also one of the main actions is finding. It is also called a determinant and is designated as follows: det. This value is determined modulo, that is, it is never negative. The easiest way to find the determinant is a 2x2 square matrix. To do this, you need to multiply the elements of the main diagonal and subtract from them the multiplied elements of the secondary diagonal.

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 match, 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, first I will 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 $A_(m\times n)=(a_(ij))$ 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 row of the matrix $A$ by the elements of the j-th column of the matrix $B$: $$c_(ij)=\sum\limits_ (p=1)^(n)a_(ip)b_(pj), \;\; i=\overline(1,m), j=\overline(1,n).$$

Let'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 a 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 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 row - 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.

1. $A+B=B+A$ (commutativity of addition)
2. $A+(B+C)=(A+B)+C$ (associativity of addition)
3. $(\alpha+\beta)\cdot A=\alpha A+\beta A$ (distributivity of multiplication by a matrix with respect to addition of numbers)
4. $\alpha\cdot(A+B)=\alpha A+\alpha B$ (distributivity of multiplication by a number relative to matrix addition)
5. $A(BC)=(AB)C$
6. $(\alpha\beta)A=\alpha(\beta A)$
7. $A\cdot (B+C)=AB+AC$, $(B+C)\cdot A=BA+CA$.
8. $A\cdot E=A$, $E\cdot A=A$, where $E$ is the identity matrix of the corresponding order.
9. $A\cdot O=O$, $O\cdot A=O$, where $O$ is a zero matrix of the appropriate size.
10. $\left(A^T \right)^T=A$
11. $(A+B)^T=A^T+B^T$
12. $(AB)^T=B^T\cdot A^T$
13. $\left(\alpha A \right)^T=\alpha A^T$

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.

Purpose of the service. Matrix calculator designed for solving matrix expressions, such as 3A-CB 2 or A -1 +B T .

Instructions. For an online solution, you need to specify a matrix expression. At the second stage, it will be necessary to clarify the dimension of the matrices.

Actions on matrices

Valid operations: multiplication (*), addition (+), subtraction (-), inverse matrix A^(-1), exponentiation (A^2, B^3), matrix transposition (A^T).

Valid operations: multiplication (*), addition (+), subtraction (-), inverse matrix A^(-1), exponentiation (A^2, B^3), matrix transposition (A^T).
To perform a list of operations, use a semicolon (;) separator. For example, to perform three operations:
a) 3A+4B
b) AB-VA
c) (A-B) -1
you will need to write it like this: 3*A+4*B;A*B-B*A;(A-B)^(-1)

A matrix is ​​a rectangular numeric table with m rows and n columns, so the matrix can be schematically represented as a rectangle.
Zero matrix (null matrix) is a matrix whose elements are all equal to zero and are denoted by 0.
Identity matrix is called a square matrix of the form

Two matrices A and B are equal, if they are the same size and their corresponding elements are equal.
Singular matrix is a matrix whose determinant is equal to zero (Δ = 0).

Let's define basic operations on matrices.

Matrix addition

Definition . The sum of two matrices of the same size is a matrix of the same dimensions, the elements of which are found according to the formula . Denoted by C = A+B.

Example 6. .
The operation of matrix addition extends to the case of any number of terms. Obviously A+0=A .
Let us emphasize once again that only matrices of the same size can be added; For matrices of different sizes, the addition operation is not defined.

Subtraction of matrices

Definition . The difference B-A of matrices B and A of the same size is a matrix C such that A+ C = B.

Matrix multiplication

Definition . The product of a matrix by a number α is a matrix obtained from A by multiplying all its elements by α, .
Definition . Let two matrices be given and , and the number of columns of A is equal to the number of rows of B. The product of A by B is a matrix whose elements are found according to the formula .
Denoted by C = A·B.
Schematically, the operation of matrix multiplication can be depicted as follows:

and the rule for calculating an element in a product:

Let us emphasize once again that the product A·B makes sense if and only if the number of columns of the first factor is equal to the number of rows of the second, and the product produces a matrix whose number of rows is equal to the number of rows of the first factor, and the number of columns is equal to the number of columns of the second. You can check the result of multiplication using a special online calculator.

Example 7. Given matrices And . Find matrices C = A·B and D = B·A.
Solution. First of all, note that the product A·B exists because the number of columns of A is equal to the number of rows of B.

Note that in the general case A·B≠B·A, i.e. the product of matrices is anticommutative.
Let's find B·A (multiplication is possible).

Example 8. Given a matrix . Find 3A 2 – 2A.
Solution.

.
; .
.
Let us note the following interesting fact.
As you know, the product of two non-zero numbers is not equal to zero. For matrices, a similar circumstance may not occur, that is, the product of non-zero matrices may turn out to be equal to the null matrix.

Solving matrices– a concept that generalizes operations on matrices. A mathematical matrix is ​​a table of elements. A table like this, which has m rows and n columns, is said to be an m by n matrix.
General view of the matrix

Main elements of the matrix:
Main diagonal. It is made up of the elements a 11, a 22…..a mn
Side diagonal. It is composed of the elements a 1n, and 2n-1.....a m1.
Before moving on to solving matrices, let’s consider the main types of matrices:
Square– in which the number of rows is equal to the number of columns (m=n)
Zero – all elements of this matrix are equal to 0.
Transposed matrix- matrix B obtained from the original matrix A by replacing rows with columns.
Single– all elements of the main diagonal are equal to 1, all others are 0.
inverse matrix- a matrix, when multiplied by which the original matrix results in the identity matrix.
The matrix can be symmetrical with respect to the main and secondary diagonals. That is, if a 12 = a 21, a 13 = a 31,….a 23 = a 32…. a m-1n = a mn-1 . then the matrix is ​​symmetrical about the main diagonal. Only square matrices are symmetrical.
Now let's move directly to the question of how to solve matrices.

Matrix addition.

Matrices can be added algebraically if they have the same dimension. To add matrix A with matrix B, you need to add the element of the first row of the first column of matrix A with the first element of the first row of matrix B, the element of the second column of the first row of matrix A with the element of the second column of the first row of matrix B, etc.
Properties of addition
A+B=B+A
(A+B)+C=A+(B+C)

Matrix multiplication.

Matrices can be multiplied if they are consistent. Matrices A and B are considered consistent if the number of columns of matrix A is equal to the number of rows of matrix B.
If A is of dimension m by n, B is of dimension n by k, then the matrix C=A*B will be of dimension m by k and will be composed of elements

Where C 11 is the sum of pairwise products of the elements of a row of matrix A and a column of matrix B, that is, the element is the sum of the product of an element of the first column of the first row of matrix A with an element of the first column of the first row of matrix B, an element of the second column of the first row of matrix A with an element of the first column of the second row matrices B, etc.
When multiplying, the order of multiplication is important. A*B is not equal to B*A.

Finding the determinant.

Any square matrix can generate a determinant or a determinant. Writes det. Or | matrix elements |
For matrices of dimension 2 by 2. Determine there is a difference between the product of the elements of the main and the elements of the secondary diagonal.

For matrices with dimensions of 3 by 3 or more. The operation of finding the determinant is more complicated.
Let's introduce the concepts:
Element minor– is the determinant of a matrix obtained from the original matrix by crossing out the row and column of the original matrix in which this element was located.
Algebraic complement element of a matrix is ​​the product of the minor of this element by -1 to the power of the sum of the row and column of the original matrix in which this element was located.
The determinant of any square matrix is ​​equal to the sum of the product of the elements of any row of the matrix and their corresponding algebraic complements.

Matrix inversion

Matrix inversion is the process of finding the inverse of a matrix, the definition of which we gave at the beginning. The inverse matrix is ​​denoted in the same way as the original one with the addition of degree -1.
Find the inverse matrix using the formula.
A -1 = A * T x (1/|A|)
Where A * T is the Transposed Matrix of Algebraic Complements.

We made examples of solving matrices in the form of a video tutorial

:

If you want to figure it out, be sure to watch it.

These are the basic operations for solving matrices. If you have additional questions about how to solve matrices, feel free to write in the comments.

If you still can’t figure it out, try contacting a specialist.

This is a concept that generalizes all possible operations performed with matrices. Mathematical matrix - table of elements. About a table where m lines and n columns, this matrix is ​​said to have the dimension m on n.

General view of the matrix:

For matrix solutions It is necessary to understand what a matrix is ​​and know its main parameters. Main elements of the matrix:

• The main diagonal, consisting of elements a 11, a 22…..a mn.
• Side diagonal consisting of elements a 1n , a 2n-1 .....a m1.

Main types of matrices:

• Square is a matrix where the number of rows = the number of columns ( m=n).
• Zero - where all matrix elements = 0.
• Transposed matrix - matrix IN, which was obtained from the original matrix A by replacing rows with columns.
• Unity - all elements of the main diagonal = 1, all others = 0.
• An inverse matrix is ​​a matrix that, when multiplied by the original matrix, results in the identity matrix.

The matrix can be symmetrical with respect to the main and secondary diagonals. That is, if a 12 =a 21, a 13 =a 31,….a 23 =a 32…. a m-1n =a mn-1, then the matrix is ​​symmetrical about the main diagonal. Only square matrices can be symmetric.

Methods for solving matrices.

Almost all matrix solving methods consist in finding its determinant n-th order and most of them are quite cumbersome. To find the determinant of the 2nd and 3rd order there are other, more rational methods.

Finding 2nd order determinants.

To calculate the determinant of a matrix A 2nd order, it is necessary to subtract the product of the elements of the secondary diagonal from the product of the elements of the main diagonal:

Methods for finding 3rd order determinants.

Below are the rules for finding the 3rd order determinant.

Simplified rule of triangle as one of matrix solving methods, can be depicted this way:

In other words, the product of elements in the first determinant that are connected by straight lines is taken with a “+” sign; Also, for the 2nd determinant, the corresponding products are taken with the “-” sign, that is, according to the following scheme:

At solving matrices using Sarrus' rule, to the right of the determinant, add the first 2 columns and the products of the corresponding elements on the main diagonal and on the diagonals that are parallel to it are taken with a “+” sign; and the products of the corresponding elements of the secondary diagonal and the diagonals that are parallel to it, with the sign “-”:

Decomposition of the determinant in a row or column when solving matrices.

The determinant is equal to the sum of the products of the elements of the row of the determinant and their algebraic complements. Usually the row/column that contains zeros is selected. The row or column along which the decomposition is carried out will be indicated by an arrow.

Reducing the determinant to triangular form when solving matrices.

At solving matrices method of reducing the determinant to a triangular form, they work like this: using the simplest transformations on rows or columns, the determinant becomes triangular in form and then its value, in accordance with the properties of the determinant, will be equal to the product of the elements that are on the main diagonal.

Laplace's theorem for solving matrices.

When solving matrices using Laplace's theorem, you need to know the theorem itself. Laplace's theorem: Let Δ - this is a determinant n-th order. We select any k rows (or columns), provided k≤ n - 1. In this case, the sum of the products of all minors k-th order contained in the selected k rows (columns), by their algebraic complements will be equal to the determinant.

Solving the inverse matrix.

Sequence of actions for inverse matrix solutions:

1. Determine whether a given matrix is ​​square. If the answer is negative, it becomes clear that there cannot be an inverse matrix for it.
2. We calculate algebraic complements.
3. We compose a union (mutual, adjoint) matrix C.
4. We compose the inverse matrix from algebraic additions: all elements of the adjoint matrix C divide by the determinant of the initial matrix. The final matrix will be the required inverse matrix relative to the given one.
5. We check the work done: multiply the initial matrix and the resulting matrix, the result should be an identity matrix.

Solving matrix systems.

For solutions of matrix systems The Gaussian method is most often used.

The Gauss method is a standard method for solving systems of linear algebraic equations (SLAEs) and it consists in the fact that variables are sequentially eliminated, i.e., with the help of elementary changes, the system of equations is brought to an equivalent system of triangular form and from it, sequentially, starting from the latter (by number), find each element of the system.

Gauss method is the most versatile and best tool for finding matrix solutions. If a system has an infinite number of solutions or the system is incompatible, then it cannot be solved using Cramer’s rule and the matrix method.

The Gauss method also implies direct (reducing the extended matrix to a stepwise form, i.e., obtaining zeros under the main diagonal) and reverse (obtaining zeros above the main diagonal of the extended matrix) moves. The forward move is the Gauss method, the reverse move is the Gauss-Jordan method. The Gauss-Jordan method differs from the Gauss method only in the sequence of eliminating variables.