Page 57 - 35Linear Algebra

2.3 Elementary Row Operations 57

• The row swap matrix that swaps the 2nd and 4th row is the identity matrix with
the 2nd and 4th row swapped:
 
1 0 0 0 0
0 0 0 1 0
 
 
0 0 1 0 0 .
 
 
 0 1 0 0 0 
0 0 0 0 1
• The scalar multiplication matrix that replaces the 3rd row with 7 times the 3rd
row is the identity matrix with 7 in the 3rd row instead of 1:

 
1 0 0 0
 0 1 0 0 
  .
0 0 7 0
 
0 0 0 1
• The row sum matrix that replaces the 4th row with the 4th row plus 9 times
the 2nd row is the identity matrix with a 9 in the 4th row, 2nd column:
 
1 0 0 0 0 0 0
 0 1 0 0 0 0 0 
 
0 0 1 0 0 0 0
 
 
0 9 0 1 0 0 0 .
 
 
 0 0 0 0 1 0 0 
 
 0 0 0 0 0 1 0 
0 0 0 0 0 0 1
We can write an explicit factorization of a matrix into EROs by keeping
track of the EROs used in getting to RREF.

Example 28 (Express M from Example 25 as a product of EROs)
Note that in the previous example one of each of the kinds of EROs is used, in the
order just given. Elimination looked like
       
0 1 1 2 0 0 1 0 0 1 0 0
M =  2 0 0  E 1  0 1 1  E 2  0 1 1  E 3  0 1 0  = I ,
∼
∼
∼
0 0 1 0 0 1 0 0 1 0 0 1
where the EROs matrices are
   1   
0 1 0 0 0 1 0 0
2
E 1 =  1 0 0  , E 2 =  0 1 0  , E 3 =  0 1 −1  .
0 0 1 0 0 1 0 0 1

57