solve x+y+z=6,x-y+z=2,2x-y+3z=9 using rank method
Answers
Step-by-step explanation:
Step 1:
The given system of equation is of the form AX=B.
(i. e) 1 2 1
1 0 3
2 −3 0 x
y
z = 7
11
1
Where A = 1 2 1
1 0 3
2 −3 0 , X = x
y
z and
B = 7
11
1
Let us now find the determinant value of A
|A|= 1 2 1
1 0 3
2 −3 0
= 1(0 + 9) − 2(0 − 6) + 1( − 3 − 0)
= 9 + 12 − 3 = 18 ≠ 0.
It is non-singular. it inverse exists.
Step 2:
Next let us find the adjoint of A
A11 = ( − 1) 1+1 0 3
−3 0 =9.
A12 = ( − 1) 1+2 1 3
2 0 =6.
A13 = ( − 1) 1+3 1 0
2 −3 =-3.
A21 = ( − 1) 2+1 2 1
−3 0 =-3.
A22 = ( − 1) 2+2 1 1
2 0 =-2.
A23 = ( − 1) 2+3 1 2
2 −3 =7.
A31 = ( − 1) 3+1 2 1
0 3 =6.
A32 = ( − 1) 3+2 1 1
1 3 =-2.
A33 = ( − 1) 3+3 0 3
−3 0 =-2.
Hence the adjoint of A is
A 11 A21 A31
A 12 A22 A32
A 13 A23 A33
= 9 −3 6
6 −2 −2
−3 7 −2
A −1 = 1
18 9 −3 6
6 −2 −2
−3 7 −2
Step 3:
A −1 B = X,substituting for A −1 ,B and X we get
x
y
z = 1
18 9 −3 6
6 −2 −2
−3 7 −2 7
11
1
= 1
18 63 − 33 + 6
42 − 22 − 2
−21 + 77 − 2 = 36
18
18
18
54
18 = 2
1
3
x
y
z = 2
1
3
x=2,y=1,z=3.