Solve the following equations by method of inversion,
x - y + z = 0, 2x + y - 3z=0, x + y + z = 0.
Answers
Answer:
for example
Solve the system of equations, using matrix method
x−y+z=4,2x+y−3z=0,x+y+z=2
Answer
Given system of equations
x−y+z=4
2x+y−3z=0
x+y+z=2
This can be written as
AX=B
where A=
⎣
⎢
⎢
⎡
1
2
1
−1
1
1
1
−3
1
⎦
⎥
⎥
⎤
,X=
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
,B=
⎣
⎢
⎢
⎡
4
0
2
⎦
⎥
⎥
⎤
Here,
∣A∣=1(1+3)+1(2+3)+1(2−1)
⇒∣A∣=4+5+1=10
Since, ∣A∣
=0
Hence, the system of equations is consistent and has a unique solution given by X==A
−1
B
A
−1
=
∣A∣
adjA
and adjA=C
T
C
11
=(−1)
1+1
∣
∣
∣
∣
∣
∣
1
1
−3
1
∣
∣
∣
∣
∣
∣
⇒C
11
=1+3=4
C
12
=(−1)
1+2
∣
∣
∣
∣
∣
∣
2
1
−3
1
∣
∣
∣
∣
∣
∣
⇒C
12
=−(2+3)=−5
C
13
=(−1)
1+3
∣
∣
∣
∣
∣
∣
2
1
1
1
∣
∣
∣
∣
∣
∣
⇒C
13
=2−1=1
C
21
=(−1)
2+1
∣
∣
∣
∣
∣
∣
−1
1
1
1
∣
∣
∣
∣
∣
∣
⇒C
21
=−(−1−1)=2
C
22
=(−1)
2+2
∣
∣
∣
∣
∣
∣
1
1
1
1
∣
∣
∣
∣
∣
∣
⇒C
22
=1−1=0
C
23
=(−1)
2+3
∣
∣
∣
∣
∣
∣
1
1
−1
1
∣
∣
∣
∣
∣
∣
⇒C
23
=−(1+1)=−2
C
31
=(−1)
3+1
∣
∣
∣
∣
∣
∣
−1
1
1
−3
∣
∣
∣
∣
∣
∣
⇒C
31
=3−1=2
C
32
=(−1)
3+2
∣
∣
∣
∣
∣
∣
1
2
1
−3
∣
∣
∣
∣
∣
∣
⇒C
32
=−(−3−2)=5
C
33
=(−1)
3+3
∣
∣
∣
∣
∣
∣
1
2
−1
1
∣
∣
∣
∣
∣
∣
⇒C
33
=1+2=3
Hence, the co-factor matrix is C=
⎣
⎢
⎢
⎡
4
2
2
−5
0
5
1
−2
3
⎦
⎥
⎥
⎤
⇒adjA=C
T
=
⎣
⎢
⎢
⎡
4
−5
1
2
0
−2
2
5
3
⎦
⎥
⎥
⎤
⇒A
−1
=
∣A∣
adjA
=
10
1
⎣
⎢
⎢
⎡
4
−5
1
2
0
−2
2
5
3
⎦
⎥
⎥
⎤
Solution is given by
X=A
−1
B
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
10
1
⎣
⎢
⎢
⎡
4
−5
1
2
0
−2
2
5
3
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
4
0
2
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
10
1
⎣
⎢
⎢
⎡
16+4
−20+10
4+6
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
10
1
⎣
⎢
⎢
⎡
20
−10
10
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
2
−1
1
⎦
⎥
⎥
⎤
Hence, x=2,y=−1,z=1