Addition

• Given matrices
  
  

• is the matrix whose entries are defined by
  
  

• Only matrices of same sizes are addible

Scalar multiplication

• Given an matrix and a number
  
  

• is the matrix whose entries are defined by
  
  

• scalar multiplication=multiply every entry by the number
  
  

Multiplication

• given an matrix and an matrix
  
  

• let be an arbitrary vector, then is an vector, and is an vector
  
  

• is the matrix that represents the linear map
  
  

• the dimension r in the middle must align

Example

Exercise: find following products

Exercise

• 
  
  

• find , , ,
  
  

• find

The identity matrix

• An matrix that is filled with on the main diagonal and elsewhere, denoted by
  
  

• is the multiplicative identity, i.e., holds for any matrix
  
  

• The additive identiy is the zero matrix (the matrix filled with 0)

Powers

• Given an matrix , and a natural number , define
  , and
  
  

• Only square matrices can be powered
  
  

• If , and the series is convergent for all , then we may write , example：

Exercise

• 
  
  

• compute ,
  
  

• compute

Summary

• Addition and scalar multiplication
  
  
• Multiplication
  
  
• Algebraic Laws