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Linear Algebra: matrix review

2022-01-15 02:11:53 YYY speaker

1. System of linear equations :

The constant terms are 0 Namely :n A system of homogeneous linear equations ( One didn't do it 0, Is a system of nonhomogeneous linear equations )
The coefficients in front of the unknowns can · form :n Order matrix

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2. matrix :m That's ok n Column : from m*n Composed of a number table :

The number of rows and columns are n: be called n Order matrix or n Square matrix

3. Matrix operation :

A number multiplied by a matrix is equal to this number multiplied by each element of the matrix

Matrix times matrix :( The number of columns of the first matrix is equal to the number of rows of the second matrix )
sm mx

Multiply a row by a column ( Multiply and add the corresponding elements ) An element that makes up the new matrix
2 That's ok 3 Column multiply 3 That's ok 2 Column become :4 Elements ( A matrix of two rows and two columns )

The sign of the matrix is :A,B,C

Inverse matrix : It's a matrix * The other matrix is equal to E( Unit matrix )
Unit matrix : Is that every element except the diagonal element is 1 The other elements are 0;
AB=E.
B Namely A The inverse matrix . Symbol :A Superscript is -1.
Inverse matrix A Equal to determinant A Multiply the reciprocal of the adjoint matrix A.

Adjoint matrix A: It's the determinant A A new matrix consisting of the algebraic cofactor values of each element

The location here :
The columns of a matrix are composed of rows of a determinant Subscript of element
The rows of a matrix are composed of columns of a determinant

Therefore, in the determinant, the algebraic cofactor values of each element are composed of a number table , Turn it upside down ( Is the adjoint matrix )
Symbol :A*

4. How to solve n A system of linear equations ?

Using Kramer's law :
n The coefficients of a system of linear equations : A determinant consisting of the coefficients of unknowns A It's not equal to 0
The equation has a unique solution :
The first unknown is equal to the determinant A Multiply the reciprocal of the value by A1.
A1: It's a determinant , It is the element of the first column that replaces the resulting matrix with a constant term A1.
This A1 The corresponding determinant A1.

The second unknown is the same

Exercise questions 1:

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Inverse matrix method to do this problem :
|A| It's not equal to 0, therefore A reversible ,
Matrix of unknowns = Inverse matrix of matrix with unknown coefficients * Constant term matrix
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