Rank of a Matrix





In this page rank of a matrix we are going to see how to calculate rank of any matrix with examples.

To find rank of any given matrix first we have to find the echelon form(triangular form)

Procedure to find Echelon form (triangular form)

(i) The first element of every non-zero row is 1.

(ii) The row which is having every element zero should be below the non zero row.

(iii) Number of zeroes in the next non zero row should be more than the number of zeroes in the previous non zero row.

Example 1:

 
1 1 -1
3 -2 3
2 -3 4
 


Solution:

˜
 
1 1 -1
3 -2 3
2 -3 4
 

R₂ => R₂ - 3R₁

        3        -2         3

         3         3        -3

        (-)       (-)       (+)

      ________________

       0         - 5        6

      ________________

R₃ => R₃ - 2R₁

       2        -3         4

       2         2        -2

      (-)       (-)       (+)

      ________________

       0         -5       6   

      ________________

˜
 
1 1 -1
0 -5 6
0 -5 6
 


R₂ => R₂ - 3R₁

R₃ => R₃ - 2R₁

R₃ => R₃ - R₂

       0        -5         6

       0        -5         6

      (-)       (+)       (-)

     ___________________

       0         0        0   

      __________________

rank of a matrix
˜
 
1 1 -1
0 -5 6
0 0 0
 


R₃ => R₃ - R₂

Number of non zero rows is 2. So rank of the given matrix = 2.

Example 2:

 
4 3 6 25
1 5 7 13
2 9 1 1
 


Solution:

˜
 
4 3 6 25
1 5 7 13
2 9 1 1
 

R₂ <-> R₁
˜
 
1 5 7 13
4 3 6 25
2 9 1 1
 

R₂ => R₂ - 4R₁

         4         3         6        25

         4         20       28       52

        (-)        (-)       (-)       (-)

      ____________________________

       0        -17      -22      -27

      ___________________________

R₃ => R₃ - 2R₁

       2          9         1         1

         2         10       14       26

        (-)       (-)       (-)       (-)

      ___________________________

       0        -1       -13      -25

      ___________________________


˜
 
1 5 7 13
0 -17 -22 -27
0 -1 -13 -25
 


R₂ => R₂ - 4 R₁

R₃ => R₃ - 2 R₁

R₃ => 17R₃ - R₂

       0        -17        -221        -425

       0        -17         -22           -27

       (-)       (+)       (+)             (+)

      _________________________________

       0         0         -199         -398    

      ________________________________


˜
 
1 5 7 13
0 -17 -22 -27
0 0 -199 -398
 



R₃ => 17 R₃ - R₂

Number of non zero rows is 2. So rank of the given matrix = 2.Now you can try the following questions to understand this topic much better.











Questions

Solution

1) Find the rank of the following matrix
 
2 1 1 5
1 1 1 4
1 -1 2 1
 

Solution

2) Find the rank of the following matrix
 
1 2 1 7
2 -1 2 4
1 1 -2 -1
 

Solution

3) Find the rank of the following matrix
 
2 5 7 52
1 1 1 9
2 1 -1 0
 

Solution

4) Find the rank of the following matrix
 
3 1 -1 2
2 -1 2 6
2 1 -2 -2
 

Solution

5) Find the rank of the following matrix
 
2 -1 3 9
1 1 1 6
1 -1 1 2
 

rank of a matrix rank of a matrix

Solution






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