The intersection of rows and columns is called an element

The elements of a matrix can be numbers, symbols, or expressions

The elements of a matrix are enclosed in parentheses

Matrix rows and columns separated by commas or semicolons

The addition of two matrices is to add the elements at corresponding positions

The subtraction of two matrices is to subtract the elements at corresponding positions.

Matrix multiplication is multiplying each row of the first matrix by each column of the second matrix

The transpose of a matrix is ​​to swap the rows and columns of the matrix

Matrices can be used to solve systems of linear equations

Matrices can be used to represent vector spaces

Matrix factorization can decompose a matrix into a simpler form

The inverse of a matrix is ​​a matrix such that multiplying the matrix by its inverse gives the identity matrix;

Definition of elementary transformation

Properties of elementary transformations

Applications of elementary transformations

Definition of elementary matrix

Properties of elementary matrices

Applications of elementary matrices

The concept of inverse matrix

Calculation of inverse matrix

Solve a system of linear equations

Rank of the matrix

Uniqueness of inverse matrix

rank of inverse matrix

The determinant is the sum of the products of all the rows or columns of a square matrix

Calculate using the properties and formulas of determinants

A determinant is equal to its transposed determinant

The determinant multiplied by its inverse determinant equals the identity matrix

The determinant is equal to the determinant of its adjoint matrix

Solve a system of linear equations

Calculate the rank of a matrix

Determine whether the matrix is ​​invertible;

Linear independence: means that any two vectors in a set of vectors are not proportional, that is, no vector is a multiple of other vectors.

Maximum linearly independent rows or columns: refers to the existence of a maximum submatrix in the matrix in which all rows or columns are linearly independent

Determinant: The determinant of a matrix is ​​the value of the determinant composed of the row or column vectors of the matrix

Rank matrix: A rank matrix is ​​a matrix composed of row or column vectors of a matrix

Elementary transformation method: convert the matrix into a row echelon matrix through elementary transformation, and then calculate the rank of the matrix based on the number of non-zero rows