Examples

• there are some some rabbits and chicken in the barn. What will be their respective numbers if there are 94 legs and 35 heads in total?
  
  

• eleminate first the bottom left, then the top right:
  
  

Examples

• 
  
  

• swap the first two rows, then eliminate
  

Diagonal Matrices

• diagonal matrix：a matrix filled with except on the main diagonal, example：
  
  

• left/right multiplying a diagonal matrix corresponds to row/column scaling, example:
  
  

Triangular Matrices

• upper/lower trangular matrix: a matrix filled with below/above the main daigonal
  example:
  
  

• strictly upper/lower trangular matrix: a matrix filled with below/above and including the main daigonal
  example:

Inverses and Products

• the inverse of a diagonal matrix is still diagonal
  
  

• the inverse of an upper/lower triangular matrix is still upper/lower triangular
  
  

• the product of an upper/lower triangular matrices is still upper/lower triangular
  
  

Elementary Row Operation I

• swap two rows
  
  

• let be the matrix obtained by swapping the -th and -th rows in
  
  

• then is the matrix obtained by swapping the -th and -th rows in
  
  

Elementary Row Operation II

• multiply the -th row by
  
  

• let be the matrix obtain by multiplying the -th row in by
  
  

• then is the matrix obtain by multiplying the -th row in by
  
  

Elementary Row Operation III

• adding a multiple of one row to anther row
  
  

• let be the matrix obtained by adding an mutliple of the -th row to the -th row in
  
  

• then is the matrix obtained by performing the same operation on

Inversion by elimination

• put and together side by side
  
  

• apply elementary row operations on both simultaneously
  
  

• eleminate to get
  
  

• then the transformed part is

Examples

Summary

• Gaussian elimination
  
  

• elementary row operations and their matrix forms
  
  

• computation of the inverse using eliminations