Rank Method in Matrix





In this page rank method in matrix we are going to see how to solve the given equations by using this method.

Procedure to find Rank method

(i) First we have to write the given equations in the form of AX = B.

(ii) Then we have to write augmented matrix [A,B].

(iii) Then we have to find rank-of-matrices A and [A,B] by applying elementary row operations.

(iv) If rank (A) = rank of [A,B] = number of unknowns then we can say that the system is consistent and it has unique solution.

(v) If rank (A) = rank of [A,B] < number of unknowns then we can say that the system is consistent and it has infinitely many solution.

(vi) If rank (A) ≠ rank of [A,B] then we can say that the system is not consistent and it has no solution.

Example 1:

Solve the following linear equation by rank-method

4x + 3y + 6z = 25

x + 5y + 7z = 13

2x + 9y + z = 1

Solution:

 
4 3 6 25
1 5 7 13
2 9 1 1
 


[A,B]

˜
 
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 - 4R₁

R₃ => R₃ - 2R₁

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 => 17R₃ - R₂

Rank (A) = 3

Rank [A,B] = 3

x + 5y + 7z = 13   --------(1)

-17y - 22z = -27  --------(2)

       -199z = -398 --------(3)

             z = -398/(-199)

             z = 2

apply z = 2 in the second equation

   -17y - 22 (2) = -27

    -17y - 44 = -27

           -17y = -27 + 44

           -17y = 17

               y = 17/(-17)

               y = -1

apply z = 2 and y = -1 in the first equation to get the value of x

 x + 5 (-1) + 7 (2) = 13

   x - 5 + 14 = 13

        x + 9 = 13

             x = 13 - 9

             x = 4

Answer :

 x = 4

 y = -1

 z = 2


Questions



Solution


1) Find the following linear equations by using rank method of matrix

2x + y + z = 5

x + y + z = 4

x - y + 2z = 1

Solution

2) Find the following linear equations by using rank method of matrix

x + 2y + z = 7

2x - y + 2z = 4

x + y - 2z = -1

Solution

3) Find the following linear equations by using rank method of matrix

2x + 5y + 7z = 52

x + y + z = 9

2x + y - z = 0

Solution

4) Find the following linear equations by using rank method of matrix

3x + y - z = 2

2x - y + 2z = 6

2x + y - 2z = -2

Solution

5) Find the following linear equations by using rank method of matrix

2x - y + 3z = 9

x + y + z = 6

x - y + z = 2

rank method in matrix rank method in matrix rank method in matrix rank method in matrix

Solution






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