How to solve a system of equations with a determinant, please give an example, thank you! 5

a11x+a12y=b1

a21x+a22y=b2

then x=b1 a12|

b2 a22|

a11 a12|

a21 a22|

A determinant is a function that defines a domain.

for the matrix a, the value range.

is a scalar quantity.

Write det(a). Essentially, a determinant describes the "volume" of a "parallelopolyhedron" formed by a linear transformation in n-dimensional space. Determinants have important applications in both calculus (e.g., commutation integral) and algebra.

The concept of determinant was first introduced in the process of solving systems of linear equations. Determinants are used to determine the number of solutions to a system of linear equations, as well as the form. Subsequently, the determinant gradually showed important significance and role in many fields.

Thus there are definitions of determinants of linear automorphism and vector groups.

The property of the determinant can be summarized as an n-order alternating linear form, which reflects the nature of the determinant as a function describing "volume".

A matrix of numbers resembles a matrix, which is represented in parentheses.

The determinant uses line segments. The value of the determinant is the algebraic sum of all the different products that can be obtained in the following way, which is a real number: each product is taken from each row in turn, and each of these factors needs to be taken from a different column, and as a multiplier, the sign of the product is precisely negative and determines whether the number of transpositions required to restore the order of the indicators of the columns of each multiplier to the natural order is even or odd.

It can also be explained in this way: the determinant is the algebraic sum of the product of all the elements of the different rows and columns of the matrix, and the sign of each term in the sum is determined by the sum of the row indicators of each element of the product and the inverse ordinal number of the column indicators: if the sum of the inverse ordinal numbers is even, the term is positive; If the sum of the inverse ordinal numbers is odd, the term is negative.

The premise of solving a system of linear equations with a determinant, i.e. Crammer's law, is as follows:

1.The number of linear equations ax=b is the same as the number of unknowns, i.e., the coefficient matrix a is a square matrix.

2.The determinant of the coefficient matrix a |a| ≠0.

Then the system of equations has a unique solution: xi = di d

d=|a|di is the determinant obtained by replacing the ith column in d with b.

Example: A system of equations.

x + 2y = 3

4x + 5y = 6d=1 2

d1=3 2

d2=1 3

So x = d1 d = -1, y=d2 d = 2

Solve a system of linear equations with a determinant, known as Crammer's law.

The prerequisites for using it are:

1.The number of linear equations ax=b is the same as the number of unknowns, i.e., the coefficient matrix a is a square matrix.

2.The determinant of the coefficient matrix a |a| ≠0.

Then the system of equations has a unique solution: xi = di d

d=|a|di is the determinant obtained by replacing the ith column in d with b.

Example: Equation Laughing Group.

x + 2y = 3

4x + 5y = 6d=

d1=d2=

So x = d1 d = -1, y=d2 d = 2

<> method of solving a system of linear equations with a determinant is called the Kramer rule.

1.The number of linear equations ax=b is the same as the number of unknowns, i.e., the coefficient matrix a is a square matrix.

2.The determinant of the coefficient matrix a |a| ≠0.

Then the system of equations has a unique solution: xi = di d

d=|a|di is the determinant obtained by replacing the ith column in d with b.

Example: A system of equations.

1 Introduction. For a system of binary linear equations.

The above solution can also be written.

Namely. Vandermund determinant (to be added).

Gramer of this brother's law (cramer).

ax=b There is always a solution to a system of homogeneous linear equations (zero solution) x=0: the constant term ax=0;

1.When|a|It is not equal to 0, only zero solution; (linear independent, full rank) 2When|a|is equal to 0 and has a non-zero solution; A system of non-homogeneous linear equations (linearly correlated).

1.When|a|Not equal to 0, the only solution.

Laplace's theorem:

ax=b is transformed into a stepped formula by elementary transformation (r is the number of rows, n is the number of columns), and when r=n (the number of equations = the number of unknowns), there is a unique solution;

When r to be continued ...

See analysis for details; Test question analysis: first calculate d, dx, dy according to the coefficients and constant terms of x, y in the system of equations, and the following is a value classification and discussion: (1) when a≠-1, a≠1, (2) when a=-1, (3) when a=1, the solution of the system of equations can be solved

Question analysis: <> 3 points.

1) When <>

, <> system of equations has a unique solution, <>

5 points. (2) When <>

, the system of <> equations has no solution; 6 points.

3) When <>

, <> system of equations has an infinite number of solutions, <

8 points.

2.Augmentation matrix (a, b) =

The elementary row is transformed to.

The elementary row is transformed to.

The elementary row is transformed to.

The elementary row is transformed to.

The elementary row is transformed to.

This results in x1 = 3, x2 = -4, x3 = -1, x4 = 1