1. Matrix solution of binary linear equations and second-order determinant

2. The numerical value of the third-order determinant

3. Full Permutation and commutation

about n Elements , Specify a standard order （ It is often stipulated to arrange from small to large ）, For an arrangement , When the order of a pair of elements is different from that specified in the standard , It constitutes a reverse order , The total number of all reverse orders in an arrangement is called the reverse order number of the arrangement .

4. exchange

A theorem ： Any two elements in an arrangement are swapped , Parity change of the number in reverse order .

inference ： p. （ accidentally ） The number of standard permutations is the number of pairs of odd permutations （ accidentally ） Count

5.n Step determinant ： Make the table in different rows and columns n The product of numbers , And prefixed with symbols , among t by p1,p2,...,p3... About natural numbers 1,2,3... The reverse order number of this arrangement , Get the value of the determinant

6. The nature of determinants

6.1 The determinant is equal to its transposed determinant

6.2 For the two lines of a newline column （ Column ）, The determinant changes sign

6.2 inference ： If the determinant has two rows （ Column ） Exactly the same , Then this determinant is equal to zero

6.3 A row of a determinant （ Column ） All elements in the are multiplied by the same number k, It's equal to using numbers k Multiply this determinant

6.3 inference ： A row in a determinant （ Column ） The common factor of all elements of can be mentioned outside the determinant notation

6.4 If a determinant has two rows （ Column ） Elements are proportional , Then this determinant is equal to zero

6.5 If a row of the determinant （ Column ） The element of is the sum of the majority , It can be divided into multiple determinants and added

6.6 Put a row of determinant （ Column ） Multiply the elements of by the same number and add to another line （ Column ） On the corresponding element , The determinant remains unchanged

￥￥7 Reduction of determinant

1 Through the rules of knowledge point 6, the determinant can be simplified into triangular determinant , Multiply the diagonal elements directly to get the result .

The characteristics of determinant simplification

2 Use the lower order determinant to express the higher order determinant

Concept ：

The remainder formula ： stay n In order determinant , hold （i,j） The second place where yuan is located i Xing He j After the column is crossed out , Left behind n-1 The order determinant is called （i,j） Cofactor of element , Write it down as

Algebraic cofactor ：

Theorem ： A determinant is equal to any of its rows （ Column ） The sum of the products of the elements of and their corresponding algebraic cofactors （ It is often possible to simplify the determinant to a row or column with only one non-zero element, and then reduce the order through the theorem ）

Special determinant ： Vandermonde determinant

inference ： Determinant a row （ Column ） Element with another line （ Column ） The sum of the products of the algebraic cofactors of the corresponding elements of is equal to zero .