Follow teacher Li to learn line generation determinant (continuous update)

Concept

• determinant ： Determinant is the algebraic sum of the products of different row and column elements （ common $n!$ term ）.
• array ： from $1,2,\dots ,n$ An ordered array of components is called a $n$ Rank arrangement , Usually use $j_1,j_2\dots j_n$ Express $n$ Rank arrangement .
• Reverse order number ： In the arrangement , If a large number is in front of a small number , Say that these two numbers form a reverse order . The total number of a permutation in reverse order is called the reverse number of this permutation , use $C(j_1{j}_2\dots j_n)$ Indicates arrangement $j_1{j}_2\dots j_n$ In reverse order . If the reverse order number of an arrangement is even , Then this arrangement is called an even arrangement, otherwise it is called an odd arrangement .
• 2 Step determinant ：$$\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$$
• 3 Step determinant ：$$\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|=a_1{b}_2{c}_3+a_2{b}_3{c}_1+a_3{b}_1{c}_2-a_3{b}_2{c}_1-a_2{b}_1{c}_3-a_1{b}_3{c}_2$$
• $n$ Step determinant ：$$\left|\begin{array}{ccccc}{a}_{11}& a_{12}& \dots & a_{1n}& \\ a_{21}& a_{22}& \dots & a_{2n}& \\ ⋮& ⋮& & ⋮& \\ a_{n1}& a_{n2}& \dots & a_{nn}& \end{array}\right|=\sum _{j_1{j}_2\dots j_n}(-1{)}^{C(j_1{j}_2\dots j_n)}{a}_{1j_1}{a}_{2j_2}\dots a_{n{j}_n}$$ Different rows and columns $n$ Algebraic sum of the product of elements , When $j_1{j}_2\dots j_n$ It is even arrangement , This item is preceded by a positive sign ; When $j_1{j}_2\dots j_n$ When it is an odd arrangement , This item is preceded by a minus sign .
• The remainder formula ： The second part of the determinant $i$ Xing He $j$ Column out , Then the remaining determinant is $a_{ij}$ The Yu Zi form of , Write it down as $M_{ij}$, remember $A_{ij}=(-1{)}^{i+j}{M}_{ij}$ by $a_{ij}$ The algebraic covalent of .

nature

• The value of the transposed determinant remains unchanged , namely $|A^T|=|A|$
• A line has a common factor $k$, The common factor $k$ When it comes to determinants , Special , All elements in a row are zero , Then the value of determinant is $0$.
• Two lines exchange determinant sign , Special , Two lines are equal , The determinant value is $0$; The two lines are proportional , The determinant value is $0$.
• All elements in a row are the sum of two numbers , Then it can be written as the sum of two determinants .$$\left|\begin{array}{ccc}{a}_1+b_1& a_2+b_2& a_3+b_3\\ c_1& c_2& c_3\\ d_1& d_2& d_3\end{array}\right|=\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ c_1& c_2& c_3\\ d_1& d_2& d_3\end{array}\right|=\left|\begin{array}{ccc}{b}_1& b_2& b_3\\ c_1& c_2& c_3\\ d_1& d_2& d_3\end{array}\right|$$
• Of a certain line $k$ Multiply to another line , The value of the determinant remains unchanged .
• Algebraic cofactor ：$$PressiOK,\; expand\; ：|A|=a_{i1}|A_{i1}|+a_{i2}|A_{i2}|+\dots +a_{in}|A_{in}|$$$$PressjColumn\; expansion\; ：|A|=a_{1j}|A_{1j}|+a_{2j}|A_{2j}|+\dots +a_{nj}|A_{nj}|$$
• The sum of the algebraic cofactor products of all elements in one row and the corresponding elements in another row is equal to $0$.$$a_{11}{A}_{21}+a_{12}{A}_{22}+a_{13}{A}_{23}=0$$

Important formula

• On （ Next ） The value of the trigonometric determinant is equal to the product of the main diagonal elements ：$$\left|\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ 0& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& & ⋮\\ 0& 0& \dots & a_{nn}\end{array}\right|=\left|\begin{array}{cccc}{a}_{11}& 0& \dots & 0\\ a_{21}& a_{22}& \dots & 0\\ ⋮& ⋮& & ⋮\\ a_{n1}& a_{n2}& \dots & a_{nn}\end{array}\right|=a_{11}{a}_{22}\dots a_{nn}$$
• On the sub diagonal determinant ：$$\left|\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & 0\\ ⋮& ⋮& & ⋮\\ a_{n1}& 0& \dots & 0\end{array}\right|=\left|\begin{array}{cccc}0& \dots & 0& a_{1n}\\ 0& \dots & a_{2,n-1}& a_{2n}\\ ⋮& & ⋮& ⋮\\ a_{n1}& \dots & a_{n,n-1}& a_{nn}\end{array}\right|=(-1{)}^{\frac{n(n-1)}{2}}{a}_{1n}{a}_{2,n-1}\dots a_{n1}$$
• Laplace expansion ：$$\left|\begin{array}{cc}A& \ast \\ O& B\end{array}\right|=\left|\begin{array}{cc}A& O\\ \ast & B\end{array}\right|=|A|\ast |B|$$$$\left|\begin{array}{cc}O& A\\ B& \ast \end{array}\right|=\left|\begin{array}{cc}\ast & A\\ B& O\end{array}\right|=(-1{)}^{nm}|A|\ast |B|$$$m,n$ Respectively $A,B$ The order of
• Vandermonde determinant ：$$\left|\begin{array}{cccc}1& 1& \dots & 1\\ x_1& x_2& \dots & x_n\\ x_1^2& x_2^2& \dots & x_n^2\\ ⋮& ⋮& & ⋮\\ x_1^{n-1}& x_2^{n-1}& \dots & x_n^{n-1}\end{array}\right|=\prod _{1\le j<i\le n}(x_i-x_j)$$
• Characteristic polynomial ： set up $A=(a_{ij})$ yes $3$ Order matrix , be $A$ Characteristic polynomials of $$\left|\begin{array}{c}\lambda E-A\end{array}\right|={\lambda }^3-(a_{11}+a_{22}+a_{33}){\lambda }^2+s_2\lambda -|A|$$ among $$s_2=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|+\left|\begin{array}{cc}{a}_{11}& a_{13}\\ a_{31}& a_{33}\end{array}\right|+\left|\begin{array}{cc}{a}_{11}& a_{22}\\ a_{32}& a_{33}\end{array}\right|$$

Kramer's law

if $n$ An equation $n$ A system of linear equations with unknown numbers ：$$\{\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n=b_1\\ a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n=b_2\\ \dots \\ a_{n1}{x}_1+a_{n2}{x}_2+\cdots +a_{nn}{x}_n=b_n\end{array}$$ The determinant of ：$$D=|A|=\left|\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& & ⋮\\ a_{n1}& a_{n2}& \dots & a_{nn}\end{array}\right|\ne 0$$ Then the equations have a unique solution ：$$x_1=\frac{D_1}{D},x_2=\frac{D_2}{D},\dots ,x_n=\frac{D_n}{D}$$ among $$D_j=\sum _{i=1}^n{b}_i{A}_{ij}=\left|\begin{array}{ccccccc}{a}_{11}& \dots & a_{1,j-1}& b_1& a_{1,j+1}& \dots & a_{1n}\\ a_{21}& \dots & a_{2,j-1}& b_2& a_{2,j+1}& \dots & a_{2n}\\ ⋮& & ⋮& ⋮& ⋮& & ⋮\\ a_{n1}& \dots & a_{n,j-1}& b_1& a_{n,j+1}& \dots & a_{nn}\end{array}\right|(j=1,2,\dots ,n)$$
If homogeneous equations ：$$\{\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n=0\\ a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n=0\\ \dots \\ a_{n1}{x}_1+a_{n2}{x}_2+\cdots +a_{nn}{x}_n=0\end{array}$$ The coefficient determinant of is not $0$, Then the equations have only zero solutions .

If homogeneous equations ：$$\{\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n=0\\ a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n=0\\ \dots \\ a_{n1}{x}_1+a_{n2}{x}_2+\cdots +a_{nn}{x}_n=0\end{array}$$ There is a nonzero solution , Then the coefficient determinant $|A|=0$.