# Linear Algebra: matrix review

2022-01-15 02:11:53

## 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

## 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：

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

https://chowdera.com/2021/12/202112122241266007.html