Write down the given system of equations in the form of a matrix equation AX = B.
Step 1 :
Find the augmented matrix [A, B] of the system of equations.
Step 2 :
Find the rank of A and rank of [A, B] by applying only elementary row operations.
Note :
Column operations should not be applied.
Step 3 :
Case 1 :
If there are n unknowns in the system of equations and
ρ(A) = ρ([A|B]) = n
then the system AX = B, is consistent and has a unique solution.
Case 2 :
If there are n unknowns in the system AX = B
ρ(A) = ρ([A| B]) < n
then the system is consistent and has infinitely many solutions and these solutions.
Case 3 :
If ρ(A) ≠ ρ([A| B])
then the system AX = B is inconsistent and has no solution.
Test for consistency and if possible, solve the following systems of equations by rank method.
Question 1 :
2x + y + z = 5
x + y + z = 4
x - y + 2z = 1
Solution :
Number of non zero rows are 3.
ρ(A) = ρ([A|B]) = 3. The system is consistent and it has unique solution.
From 1^{st} row,
x + y + z = 4 -----(1)
From 2^{nd} row,
-y-z = -3 -----(2)
From 3^{rd} row,
3z = 3 -----(3)
From (3)
z = 1
By applying the value of z in (2), we get
-y-1 = -3
-y = -3+1
-y = -2 and y = 2
By applying the value of y and z in (1), we get
x + 2 + 1 = 4
x = 4-3
x = 1
x = 1, y = 2 and z = 1
Question 2 :
x + 2y + z = 7
2x - y + 2z = 4
x + y - 2z = -1
Solution :
Number of non zero rows are 3.
ρ(A) = ρ([A|B]) = 3. The system is consistent and it has unique solution.
From the 1^{st} row,
x+2y+z = 7 ----(1)
From the 2^{nd} row,
5y = 10 ----(2)
y = 2
From the 3^{rd} row,
-15z = -30 ----(3)
z = 2
By applying the value of y and z in (1), we get
x + 2(2) + 2 = 7
x + 6 = 7
x = 7 - 6
x = 1
x = 1, y = 2 and z = 2
Question 3 :
x + 9y - z = 27
x - 8y + 16z = 10
2x + y + 15z = 37
Here ρ(A) = ρ([A|B]) = 2 < 3, then the system is consistent and it has infinitely many solution.
From the 1^{st} row,
x + 9y-z = 27 ---(1)
From the 2^{nd} row,
17y + 17z = -17 ---(2)
Dividing by 17, we get
y + z = -1
Put z = t
y = -1 - t
By applying the value of y and z in (1), we get
x + 9(-1 - t) - t = 27
x - 9 + 9t - t = 27
x = 27 + 9 - 8t
x = 36 - 8t
x = 36 - 8t, y = -1 - t and z = t where t ∈ Real numbers.
Question 4 :
x + y + z = 3
3x + 2y + z = 3
-x - y - z = 1
Solution :
Here ρ(A) = 2, ρ([A|B]) = 3
ρ(A) ≠ ρ([A|B]) then the system is inconsistent and it has no solution.
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