Linear algebra: Chapter One determinant (1) n-order determinant of the property _ linear algebra

The first section N-order determinant one. Mathematical Concepts

1. Number of reverse order

For n different elements, we first specify a standard order between the elements (for example, n a different natural number, can be specified from small to large for the standard order), so in any of these n elements in any arrangement, when a certain two elements of the order and standard order is not the same, said there are 1 reverse. The total number of all reverse order in an arrangement is called the reverse number of this permutation.

2. Odd arrangement and even arrangement

The arrangement of the odd order number is called the odd arrangement, and the order of the number of even numbers is called the even arrangement.

3. Trade

In the permutation, any two elements are exchanged, the remaining elements are not moved, and this new arrangement is called swapping. Swap adjacent two elements, called adjacent swap

4. N-Order Column

A table that defines the number of N2 and rows n rows n columns

Make a product of the number of N in the table in different rows and columns, and give the symbol (-1) Tau to get the shape

Of the items, where P1,P2,..., pn, a permutation of the natural number 1,2,..., N, the number of reverse order for this permutation. Because such permutations have a total of n!, the n! items are in the form of an upper-form item. All this n! Dai.

Called N-order determinant, recorded as

denoted as det (). The elements of a number called the determinant det (). Two Basic principle Formula

Theorem 1.1 an arrangement of any two elements in a permutation, permutation and change of parity.

The deduction of the odd permutation is odd, and the number of permutations adjusted to the standard arrangement is even.

Theorem 1.2 N-order determinant can also be defined as

where T is the number of reverse order for the row label.

Formula 1

Formula 2

Formula 3

Three Analysis of key points and difficulties

The concept of reverse number and the definition of n-order determinant are abstract and difficult to understand. But to grasp the essence and find the law, we can thoroughly understand the essence of these concepts.

About the order of a permutation of the number, it is equal to the sum of the reverse of each number, and for the reverse order, you can find the number of later in the arrangement is smaller than the number of numbers, you can also find the number of previous numbers in the arrangement is larger.

With regard to the N-order determinant, it evaluates to a number or a polynomial, and its general term is one element per row (and only one element), requiring the product of n elements that are taken on different columns. Arrange the row labels (column labels) of these n elements in a natural order, with the rows of their corresponding column labels (or row labels) listed. It is an n-level permutation of the 1,2,..., N, which consists of n numbers. The symbol of the entry is determined by the parity of the number of reverse order in which the column is marked (ROW). Sum the general item (n!), which is the value of the determinant.

It is very complicated and difficult to compute the determinant by the definition of determinant. So, we use the definition of n-order determinant to calculate three special formulas. You can use this formula to calculate the determinant in the future. But how to turn a general determinant into the form of three formulas, that is the important content of the next section, namely the determinant of the nature.

The definition of determinant is the basis of determinant calculation, and is the focus of learning determinant. Four Analysis of typical examples

Example 1. Set

，

The coefficient of the coefficient is ___; The constant term is ___.

Solution: The solution is mainly to use the definition of determinant, because it is about the unknown is a 4-time polynomial of x, and the item containing x4 only one item, with two items and a constant term of. So the coefficient is 2, the coefficient is 10, the constant term is 12.

Section II The nature of a determinant. Mathematical Concepts

Transpose determinant

Set

Determinant DT is called the transpose determinant of row-type column D. Two Basic properties

The property 1 determinant is equal to its transpose determinant.

Property 2 interchange determinant of two rows (columns), the determinant variable.

Inference if the determinant has two rows (columns) exactly the same, the determinant is zero.

Property 3 The determinant of a row (column) in which all elements are multiplied by the same number of k, equal to the number of k multiplied by this determinant.

The common factor of all elements of a row (column) in a determinant can refer to the outside of the determinant symbol.

Property 4 The determinant is equal to zero if there are two rows (column) elements proportional.

Property 5 If the elements of a row (column) of a determinant are the sum of two digits, such as the sum of the Elements of column J, which are two numbers:

Then d equals the sum of the following two determinant

Property 6 Multiply the elements of a column (row) of a determinant by the same number and then add to the corresponding element in another column (row), and the determinant is unchanged.

Three Key points and difficulties analysis

The six basic properties of a determinant are the key to the three basic formulas given in the first section of a general determinant, it is an important theory that the determinant must master, and its difficulty is how to use the six basic properties of determinant flexibly and conveniently to calculate the value of determinant. Mastering the skill of operation can improve the speed and accuracy of operation, so as to achieve a multiplier effect. Four Analysis of typical examples

We know that, given the definition of a determinant, it is more troublesome to use the definition to calculate the determinant directly, for example, to calculate the 4-order determinant, use the definition to calculate the 24 items, a n-order such as the determinant to calculate the algebraic sum of the n!, but only the six basic properties of the determinant, can be reduced, so that the determinant is quickly calculated.

Example 2. Calculation

Solution: Because all columns of the determinant are added together with the same number of A + (n-1) x, based on this feature, we use the determinant of the nature of 6, the DN of the 2nd column, column 3rd, ..., the nth column of 1 time times at the same time add to the 1th column, and then by the nature of the deduction of 3, the common factor A + (n-1) x proposed, to

Through the example above, we see that the main determinant of the nature of a determinant into a formula form, and then directly with the results of the formula. This is simpler than using definitions directly, but it takes a lot of steps to get results. Moreover, we also realized that the determinant with lower order is simpler than that of order, so we continue to explore the new method of calculating determinant.

From:http://dec3.jlu.edu.cn/webcourse/t000022/teach/index.htm