Page 59 - 35Linear Algebra

2.3 Elementary Row Operations 59

where the EROs and their inverses are

     
1 0 0 0 1 0 0 0 1 0 0 0
 0 1 0 0   0 1 0 0   0 1 0 0 
E 1 =   , E 2 =   , E 3 =  
2 0 1 0 0 0 1 0 0 0 1 0
     
0 0 0 1 0 1 0 1 0 0 −1 1
     
1 0 0 0 1 0 0 0 1 0 0 0
0 1 0 0  −1  0 1 0 0  −1  0 1 0 0 

−1
E 1 =   , E 2 =   , E 3 =   .
 −2 0 1 0   0 0 1 0   0 0 1 0 
0 0 0 1 0 −1 0 1 0 0 1 1
Applying inverse elementary matrices to both sides of the equality U = E 3 E 2 E 1 M
gives M = E −1 E −1 E −1 U or
1 2 3
      
2 0 −3 1 1 0 0 0 1 0 0 0 1 0 0 0 2 0 −3 1
 0 1 2 2   0 1 0 0  0 1 0 0  0 1 0 0  0 1 2 2 
=
      
−4 0 9 2 −2 0 1 0 0 0 1 0 0 0 1 0 0 0 3 4
      
0 −1 1 −1 0 0 0 1 0 −1 0 1 0 0 1 1 0 0 0 −3
     
1 0 0 0 1 0 0 0 2 0 −3 1
 0 1 0 0   0 1 0 0   0 1 2 2 
=      
−2 0 1 0 0 0 1 0 0 0 3 4
     
0 0 0 1 0 −1 1 1 0 0 0 −3
   
1 0 0 0 2 0 −3 1
0 1 0 0 0 1 2 2
   
=     .
−2 0 1 0 0 0 3 4
   
0 −1 1 1 0 0 0 −3
This is a lower triangular matrix times an upper triangular matrix.

59