Solve the following simultaneous equations through inverse matrix method:
z+y=z=1 2x-2y+z=6 2+3=0
Answers
Explanation:
Given
2x−y+3z=9 ………..(1)
x+y+z=6 …………..(2)
x−y+z=2 …………….(3)
using matrix inversion method
AX=B
⎣
⎢
⎢
⎡
2
1
1
−1
1
−1
3
1
1
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
9
6
2
⎦
⎥
⎥
⎤
and X=A
−1
B where A
−1
=
∣A∣
adj(A)
Now adjA=
⎣
⎢
⎢
⎡
2
0
−2
−2
−1
−1
−4
1
3
⎦
⎥
⎥
⎤
∣A∣=2(1+1)−1(1−1)+3(−1−1)
∣A∣=4−6
∣A∣=−2.
Now A
−1
=
2
−1
⎣
⎢
⎢
⎡
2
0
−2
−2
−1
−1
−4
1
3
⎦
⎥
⎥
⎤
Now X=A
−1
B
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
2
−1
⎣
⎢
⎢
⎡
2
0
−2
−2
−1
−1
−4
1
3
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
9
6
2
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
2
−1
⎣
⎢
⎢
⎡
18−12−8
0−6+2
−18−6+6
⎦
⎥
⎥
⎤
=
2
−1
⎣
⎢
⎢
⎡
−2
−4
−18
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
1
2
9
⎦
⎥
⎥
⎤
Hence [x=1,y=2 & z=9].
Given
2x−y+3z=9 ………..(1)
x+y+z=6 …………..(2)
x−y+z=2 …………….(3)
using matrix inversion method
AX=B
⎣
⎢
⎢
⎡
2
1
1
−1
1
−1
3
1
1
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
9
6
2
⎦
⎥
⎥
⎤
and X=A
−1
B where A
−1
=
∣A∣
adj(A)
Now adjA=
⎣
⎢
⎢
⎡
2
0
−2
−2
−1
−1
−4
1
3
⎦
⎥
⎥
⎤
∣A∣=2(1+1)−1(1−1)+3(−1−1)
∣A∣=4−6
∣A∣=−2.
Now A
−1
=
2
−1
⎣
⎢
⎢
⎡
2
0
−2
−2
−1
−1
−4
1
3
⎦
⎥
⎥
⎤
Now X=A
−1
B
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
2
−1
⎣
⎢
⎢
⎡
2
0
−2
−2
−1
−1
−4
1
3
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
9
6
2
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
2
−1
⎣
⎢
⎢
⎡
18−12−8
0−6+2
−18−6+6
⎦
⎥
⎥
⎤
=
2
−1
⎣
⎢
⎢
⎡
−2
−4
−18
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
1
2
9
⎦
⎥
⎥
⎤
Hence [x=1,y=2 & z=9].
