Page 94 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Let us perform row operations on A



  1 i  0 2 i  0 2 i   0 1   0 1
  (2)   (2)   (2)   (2)  

A   i 3   i 3  0 3 2i  0 3 2i  0 0  B

         
         
 1 2  1 2   1 2   1 0   1 0
Now BX = 0
gives us
 0 1

B = 0 0 
 
 
 1 0
has only the trivial solution;
x = 0
1
x = 0
2
Definition: An m × n matrix B is called row-reduced if:
(a) the first non-zero entry in each non-zero row of B is equal to 1;
(b) each column of B which contains the leading non-zero entry of some row has all its other
entries 0.

Example 2: One example of a row-reduced matrix is the n × n identity matrix I. This is the
n × n matrix defined by

 1 if i  j
I    
ij
ij
 0 if i  j
Here we have introduced Kronecker delta ().

Example 3: Find a row reduced matrix which is equivalent to

 2  1 3 2 
 
A   1 4 0  1 
 2  6  1 5  

Now
 
 2  1 3 2   0  9 3 4   0  9 3 4  0   9 3 4 
  (2)   (2)   (1)  
A  1 4 0  1  1 4 0  1  1 4 0  1   1 4 0  1 
       
 2  6  1 5    2  6  1 5    0   2  1 7   0  1 1  7 
 2 2
   15 55  11
0   9 3 4   0 0 2  2   0 0 1  3 
     
(2) (2) (1)
 1  0  2 13   1 0  2 13    1 0  2 13 

 1 7   1 7   1 7 
0  1    0 1    0 1  
 2 2  2 2   2 2 

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