In practice, there are many problems in solving n-ary linear equations, such as

①

②

Determinant can be used to solve the problem of equal N-dimensional linear equations.

2

2. 1 permutation definition 1 an ordered array composed of1.2 ... n is called horizontal permutation. The total number of n-level arrangements is

(factorial of n)

Definition 2 In an arrangement, if the front and back positions of a group of numbers are in the opposite order of size, that is, the first number is greater than the last number, they are called reverse order. The total number of inverses in an arrangement is called the number of inverses in this arrangement.

Definition 3 Even reverse order arrangement is called even order arrangement, and odd reverse order arrangement is called odd order arrangement.

2.2 determinant

Definition (set to order n): determinant of order n.

It is the algebraic sum of the products of n elements taken from different rows and columns, and consists of items, in which the items with positive and negative signs account for half respectively, indicating the reverse order number of the arrangement.

Properties of 2.3-order determinant

The determinant of attribute 1 is equal to its transposed determinant. ()

In fact, if you remember,

Note: The rows and columns in the determinant have the same state, so the attributes of the determinant are also applicable to the rows and columns.

Property 2 Interchanges two rows () or two columns () of the determinant, and the determinant changes sign.

take for example

It is inferred that if the two rows (columns) of the determinant are exactly the same, then.

Proof: if you switch the same two lines, there will be, so.

Property 3 All elements of a row (column) of a determinant are multiplied by a number, which is equal to the number multiplied by this determinant, that is,

Inference: The common factor of all elements in a row (column) in (1) can be mentioned outside the determinant symbol;

(2) If all elements in a row (column) are zero, then;

Property 4: If there are two rows (columns) of elements in the determinant, the determinant is equal to zero.

Property 5: If all elements in a row (column) of a determinant are the sum of two numbers, then this determinant is equal to the sum of two determinants. The elements in this row (column) of these two determinants are respectively one of the two corresponding addends, and the elements in other rows (columns) are the same as the original determinant. namely

.

Syndrome: defined by determinant

Property 6 All elements of a certain row (column) of the determinant are multiplied by the same number and added to the corresponding elements of another row (column), and the value of the determinant remains unchanged, that is,

Common method for calculating determinant: use property 2, 3, 6, especially property 6 to convert determinant into upper (lower) triangular determinant, so that the value of determinant is easy to calculate.

2.4 Calculation of Determinant

2.4. Determinant calculation of1number

1. Triangle method

Example 1.

Solution: The characteristic of this determinant is that the sum of elements in each row (column) is equal. According to the nature of determinant, the second, third, fourth, fifth, fifth, fifth, fifth, sixth, sixth, sixth, seventh, seventh, eighth, ninth, ninth and ninth.

.

Example 2.

Solution: This is a low-order numerical determinant, which is usually calculated as an upper (lower) triangular determinant.

2.

2. Recursive method

Example 3 Calculate the value of determinant.

The solution adds all the columns to the 1 column, and expands them according to the 1 column to get the recursive formula.

Continue to use this recursive formula, there are

And initial value, so

Example 4 Calculation.

Solution:

.

, ,

,

3. Mathematical induction

When sum is the determinant of the same type, mathematical induction can be considered. Generally, the guess value of determinant is obtained by incomplete induction, and then the proof of guess is given by mathematical induction. Therefore, mathematical induction is generally used to prove determinant equation.

Example 5 Calculate determinant.

Solution: Combining the properties of determinant and the law of quadratic determinant, this determinant can be solved by mathematical induction.

When,

Suppose, yes

Then, when according to the first column, we get

Therefore, for any positive integer, there are

4. Formula method

Example 6 Calculate the value of determinant.

In order to solve this problem, the determinant multiplication formula is used to obtain

Because the coefficient is+1, so.

2.4.2 Examples of the concept and properties of determinant

Example 7 is known as one of the determinant of order 6. Try to determine the value and symbol of this item.

According to the definition of determinant, the solution is the algebraic sum of the products of different row and column elements. Therefore, the line indicator

It should be taken from the arrangement of 1 to 6, so you can also look at it this way.

Directly calculate the number of rows and columns in reverse order, both of which are available.

I also know that this item should be marked with a negative sign.

2.4.3 Calculation of abstract determinant

Example 8 If the fourth-order matrices A and B are similar, then the determinant () is the eigenvalue of matrix A. ..

The solution is from a to b, and the eigenvalue of b is known. Then the eigenvalues are 2, 3, 4 and 5. Then the eigenvalue is 1, 2, 3, 4. There is a formula.

2.4.4 Determinant calculation with parameters

Example 9 is known, find.

This solution multiplies-1 in line 3 by line 1, where

therefore ...

2.4.5 Proof about

Think about solving problems:

① Establishment of evidence method;

(2) reduction to absurdity: if a contradiction is found from a reversibility;

③ Construct homogeneous equation and try to prove that it has non-zero solution;

④ Try to prove the rank of matrix;

⑤ Prove that 0 is the eigenvalue of matrix A. ..

2.4.6 Solving Special Determinant

1 vandermonde determinant

Definition: Determinant is called N-order Vandermonde determinant.

Example 10 Calculate the value of determinant.

The solution rewrites 1 into, and the first line becomes the sum of two numbers, which can be divided into the sum of two determinants, namely

Remember that these two determinants are summed separately and then obtained by Vandermonde determinant.

therefore

Laplace theorem

Suppose that there is any row in determinant D, and the sum of the products of all subexpressions and their algebraic cofactors composed of elements in this row is equal to determinant.

(where: ① Level sub-formula: select any row and column in a level determinant. The elements located at the intersection of these rows and columns form a rank determinant in the original order, which is called the rank subformula of the determinant. (2) Complement formula: the determinant composed of the remaining elements in the original order after this row and this column are crossed out is called the complement formula of the complement formula. (3) Algebraic cofactor: the row and column indicators of the set-level subfactor in are the algebraic cofactor after the complement of the rule is signed).

Example 1 1 Find the determinant.

Solution: Take the first line and the second line in the determinant and get six sub-formulas:

Their corresponding algebraic cofactors are

According to Laplace theorem

3 Conclusion

The teacher's profound knowledge, keen thinking and democratic and rigorous style have benefited me a lot and I will never forget it. Strict attitude and meticulous spirit will always inspire and spur me to study hard and work hard.

Thank you for your concern, guidance and teaching!

Thank you for your concern and help.