The determinant of a matrix, determinate (referred to as det), is a scalar calculated based on the row and column data contained in the matrix. is introduced for solving systems of linear equations.
1 Definition of determinant
1.1 Second-order determinant
For a system of second-order linear equations
If both b1b2 are 0, it is called a homogeneous linear equation system, otherwise it is called an inhomogeneous linear equation system
When:
Solved by elimination method:
Then simplify to the following formula:
Then solve the above binary linear equations to get:
Diagonal rules are available for second and third order determinants.
1.2 N order determinant
Randomly select m (m≤n) elements from n different elements and arrange them in a certain order, which is called an arrangement of taking m elements from n different elements. When m=n, all permutations are called full permutations. There are n! number of permutations of order n.
For example, there are 6 kinds of full arrangements of the three elements of 1, 2, and 3, as follows:
1,2,31,3,22,1,32,3,13,1,23,2,1
In a permutation, if a larger number is arranged before the smaller number, the two numbers are said to form a reverse order. The total number of inverses in a permutation is called the inverse number of the permutation.
Use f(j1j2...jn) to represent the reverse sequence number of j1j2...jn, for example f(31542)=5
3 之前有0个数比它大1 之前有1个数比它大5 之前有0个数比它大4 之前有1个数比它大2 之前有3个数比它大0+1+0+1+3=5
Through the analysis of the second-order and third-order determinants, the formulas of the second-order, third-order and n-order can be obtained, as follows:
The following example calculation:
2 Properties of the determinant
It is very difficult to calculate the value of the determinant with the definition. Generally, the properties of the determinant are used to simplify the determinant into an upper (lower) triangle determinant before calculation.
Specifies the determinant transformation sign as follows:
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Determinant i row (column) is exchanged with j row (column), recorded as ri↔rj(ci↔cj)
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The i-th row (column) of the determinant is multiplied by the constant c, which is recorded as cri(cci)
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The k times of the j-th row (column) of the determinant is added to the i-th row (column), recorded as ri+krj(ci+kcj)
Property 1: The determinant row and column are interchanged, and its value remains unchanged
Property 2: Multiplying all elements of any row (column) of the determinant by a number k is equal to multiplying the determinant by the number k
Property 3: If each element of a certain row (column) of the determinant is 0, then the value of the determinant is changed to 0
Property 4: If each element of a row (column) of the determinant is the sum of two elements, it is equal to the sum of the two determinants
Property 5: Exchange certain two rows (columns) of a determinant, and the values of the determinant have different signs
Property 6: If two rows (columns) of a determinant are identical or proportional, the value of the determinant is zero
Property 7: Add the multiple of a certain row (column) of the determinant to another row (column), and the value of the determinant remains unchanged
The figure below is a simplified example:
3 Algebraic remainders
In the n-order determinant D, the row and column where the element aij is located is crossed out, and the remaining elements form the n-1 order determinant in the original order, which is called the remainder of the element aij, which is Mij. It is recorded that Aij=(-1)^(i+j)Mij, Aij is called the algebraic remainder of element aij.
Algebraic cofactors are mainly used to convert high-order determinants into low-order determinants.
The algebraic remainder in the following figure:
Property 1: If row i is 0 except for aij which is not 0, then the value of the determinant is as follows:
Property 2: The n-order determinant D is equal to the sum of the products of each element of any row (column) of it and its corresponding algebraic cofactor, as follows:
Property 3: The sum of the algebraic cofactor products of each element of a certain row (column) of n-order determinant D and the corresponding element of another row (column) is equal to 0, as follows:
