How did the determinant evolve?

You're right.

First, there is a solution to a system of binary linear equations.

Find the pattern of understanding.

For example, x1 = (b1a22-a12b2) (a11a22-a12a21).

The numerator denominator can be abbreviated as a determinant.

b1 a12

b2 a22

And. a11 a12

a21 a22

Similarly, x2 has similar results.

Solutions to a system of ternary linear equations.

X1 is obtained by the elimination method, and its numerator and denominator are respectively.

It is estimated that you made a mistake in your calculation to multiply the 4 numbers.

The origin of linear algebra is to solve systems of linear equations, and matrices and determinants are the tools (notations) designed in the research process. But then matrices and determinants became too powerful on their own and had wider applications.

1. Linear algebra provides a powerful mathematical tool for studying and dealing with linear problems involving many variables, and this tool has a wide range of applications in engineering technology, economic science, management science, and computer science.

2. The core content of linear algebra is to study the existence conditions of the solution of linear equations, the structure of the solution, and the method of finding the solution. The basic tool used is the matrix. And determinants are one of the most effective tools for studying matrices.

3. Find the solution of the system of linear equations according to Kramer's rule: xj=dj dxj [the jth solution, j is the jth column].

dj [replace the elements of column j, in the determinant, with constants b]d [the determinant of the coefficient term of the n-element system of linear equations]dn= (-1) i+1 ( ai1)(mi1) cannot be directly used by the diagonal rule, but dn=a11a11+a21a 21+......an1an1

The determinant is a scalar of the matrix, which is the arithmetic sum of the arrangement of the various elements in the matrix according to a certain law. There are three ways to define a determinant:

Algebraic remainder definition: According to the algebraic remainder of each element in the matrix, it is obtained according to a certain calculation rule.

The determinant is defined by row: by the first row or column of the matrix, then recursively by the cocontinuum, and finally by a numeric value.

Definition of the properties of the determinant: the interchange of different rows or columns changes the sign of the determinant, one row or column of the determinant is proportional to the linear combination of another row or column, all elements of a row or column of the determinant are multiplied by the same number k, and the value of the determinant is also multiplied by k.

Determinant algebra plays a very important role in the study of equations, inverse matrices, and calculates eigenvalues and eigentropic quantities. Therefore, it is important to master the definition and operation methods of determinants in advanced mathematics and linear algebra.

In mathematics, a determinant is a function whose domain is defined as a matrix a of det, and its value is a scalar, written as det(a) or | a |Whether it is algebra, polynomial theory, or in calculus (such as commutation integration), determinants have important applications as a basic mathematical tool.

The determinant can be seen as a generalization of the concept of directed area or volume in general Euclidean space. Or, in n-dimensional Euclidean space, the determinant describes the effect of a linear transformation on the "volume".

Chinese name. Determinant.

Foreign name. determinant (English) déterminant (French).

Expression. d=|a|=deta=det(aij)

Applied Disciplines. Linear algebra.

Scope of application.

Mathematics, Physics.

Fast. Navigation.

Quality. Mathematical definitions.

nth-order determinant.

Establish. It is composed of n2 numbers in the form of an n-order square matrix aij(i,j=1,2,..n) a number whose value is n! Sum of items.

where k1 and k2 ,..kn is to ,.. sequence 1,2n elements are exchanged k times to obtain a sequence with a sign that represents the ,.. for k1 and k2

kn takes 1, 2 ,..All permutations of n are summed, then the number d is called the corresponding determinant of the nth order square. For example, the fourth-order determinant is 4!

The shape is. where a13a21a34a42 corresponds to k=3, i.e., the sign at the front end of the term should be.

If the n-order square matrix a=(aij), then the corresponding determinant d of a is denoted as.

d=|a|=deta=det(aij)

If the corresponding determinant of matrix a is d=0, a is called a singular matrix, otherwise it is called a non-singular matrix.

Label set: Sequence 1, 2 ,..Take any k elements i1 and i2 ,.. nIK satisfied.

1≤i1i1,i2,..The whole of a subcolumn with k elements is denoted as c(n,k), and apparently c(n,k) is shared.

Subcolumns. Thus c(n,k) is a set of labels with individual elements (cf. Chapter 21, 1, ii), and the elements of c(n,k) are denoted as , and c(n,k) is denoted by .

Yes, it is a subcolumn of satisfying (1). If let = c(n,k), then = means i1=j1,i2=j2,..ik=jk。

The determinant is a matrix-like square matrix composed of several numbers of Zen beams, and unlike the matrix blind chain, the matrix is represented with brackets, while the determinant is represented by line segments.

A matrix is made up of numbers, or more generally, of certain elements.

The value of the determinant is the algebraic sum of all the different products that can be obtained in the following way, i.e., a real number.

When finding each product, one meta-factor is taken from each row in turn, and each meta-factor needs to be taken from a different column, as a multiplier, and the sign of the product is exactly negative and determines whether the number of transpositions required to restore the order of the indicators of each multiplier column to the natural order is even or odd.

It can also be explained in this way: the determinant is the algebraic sum of the product of all the elements of the different rows and columns of the matrix, and the sign of each term in the sum is determined by the sum of the row indicators of each element of the product and the inverse ordinal number of the column indicators: if the sum of the inverse ordinal numbers is even, the term is positive; If the sum of the inverse ordinal grinding is odd, the term is negative.

The definition of the determinant is calculated by the number of n numbers in the form of an n-order square matrix aij(i,j=1,2,..n) to determine a number whose value is the sum of n terms, calculated using the properties of the determinant.

The determinant is a method of calculating the determinant, let a1j, a2j ,..., Lulet anj (1 j n) be the nth order determinant d=|aij|element in any column.

And a1j, a2j ,..., anj are their algebraic coundons in d, then d=a1ja1j+a2ja2j+....+anjanj is called the determinant d of the column.

The calculation of the determinant uses the property of the determinant, and the essence of the determinant is a number, so the collapse and change of the determinant is an equal change based on the existing properties, and what changes is the "appearance" of the determinant.

A determinant is a mathematical tool used to describe certain properties of a matrix. It is usually represented by a square matrix, and its value can be calculated by performing a series of operations on the numbers in the matrix. Determinant spring dust has a wide range of applications in many fields of mathematics and science, such as linear algebra, calculus, physics, engineering, etc.

The basic nature of the determinant adds the elements of all rows to any line.

When there is a row with a determinant and all the elements of the columns add up to the same result, we want to add all the rows or all the columns together. Finally, the element "3+" in column 1 should be extracted.

Common factor. If the items of the polynomial have a common factor, you can put the common factor out of the parentheses, and write the polynomial in the form of the product of the factor, and this method of factoring is called the common factor method.

Specific method: when all coefficients are integers, the coefficients of the common factor should be taken as the greatest common divisor of each coefficient; The letters are the same letters, and the index of each letter is the lowest number; Take the polynomial of the same and the number of polynomials is the lowest.

In mathematics, a determinant is a function that defines the rubber field as a matrix a of det, and the hall guess value as a scalar, written det(a) or | a |The determinant can be seen as the generalization of the concept of directed area or volume in general Euclidean space.