## 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 = -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  =  4

Solution :

x  =  4,  y  =  -1 and  z  =  2

## Practice Questions in Finding Rank Method

Find the following linear equations by using rank method of matrix.

(i)  2x + y + z  =  5, x + y + z  =  4, x - y + 2z  =  1

Solution

(ii)  x + 2y + z  =  7, 2x - y + 2z  =  4, x + y - 2z  =  -1

Solution

(iii)  2x + 5y + 7z  =  52, x + y + z  =  9, 2x + y - z  =  0

Solution

(iv)  3x + y - z  =  2, 2x - y + 2z  =  6, 2x + y - 2z  =  -2

Solution

(v)  2x - y + 3z  =  9, x + y + z  =  6, x - y + z  =  2

Solution

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