- Solve the following system of equations using matrix inversion méthod: x + 2y - 3z=6, 3x+2y-2z= 3, 2x-y+z= 2
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Explanation:
Given system of equations is
3x+2y−2z=3
x+2y+3z=6
and 2x−y+z=2
In the form of AX=B
=
⎣
⎢
⎢
⎡
3
1
2
2
2
−1
−2
3
−1
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
3
6
2
⎦
⎥
⎥
⎤
For A
−1
,∣A∣=∣3(5)−2(1−6)+(−2)(−5)∣
=∣15+10+10∣=∣35∣
=0
∴A
11
=5,A
12
=5,A
13
=−5
A
21
=0,A
22
=7,A
23
=7
A
31
=10,A
32
=−11,A
33
=4
∴adjA=
∣
∣
∣
∣
∣
∣
∣
∣
5
0
10
5
7
−11
−5
7
4
∣
∣
∣
∣
∣
∣
∣
∣
T
=
∣
∣
∣
∣
∣
∣
∣
∣
5
5
−5
0
7
7
10
−11
4
∣
∣
∣
∣
∣
∣
∣
∣
Now A
−1
=
∣A∣
adjA
=
35
1
∣
∣
∣
∣
∣
∣
∣
∣
5
5
−5
0
7
7
10
−11
4
∣
∣
∣
∣
∣
∣
∣
∣
For X=A
−1
B.
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
35
1
∣
∣
∣
∣
∣
∣
∣
∣
5
5
−5
0
7
7
10
−11
4
∣
∣
∣
∣
∣
∣
∣
∣
=
35
1
⎣
⎢
⎢
⎡
35
35
35
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
1
1
1
⎦
⎥
⎥
⎤
∴x=1,y=1 and z=1
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