4. The prices of three commodities A Band Carey and tz per unit respectively p purchases 4 units of C and sells 3 units of A and 5 units of B. O parchases 3 units of Band sells Zanits of A and I unit of C. R purchases 1 unit Of A and sells 4 units of Band 6 units of C. In the process P. Qand Rearn 26,000 25.000 and 213,000 respectively. By using matris inversion method, find the prices per unit of A, B and C.
Answers
Answer:
For A:3x+5y−4z=6000
For B:2x−3y+z=5000
For C:−x+4y+6z=13000
These equations can be written as AX=B.
where A=
⎣
⎢
⎢
⎡
3
2
−1
5
−3
4
−4
1
6
⎦
⎥
⎥
⎤
,
X=
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
and A=
⎣
⎢
⎢
⎡
6000
5000
13000
⎦
⎥
⎥
⎤
∣A∣=3(−18−4)−5(12+1)−4(8−3)=−66−65−20=−151
=0
Therefore A is nonsingular matrix and inverse exists.
A
11
=(−1)
1+1
∣
∣
∣
∣
∣
∣
−3
4
1
6
∣
∣
∣
∣
∣
∣
=−18−4=−22
A
12
=(−1)
1+2
∣
∣
∣
∣
∣
∣
2
−1
1
6
∣
∣
∣
∣
∣
∣
=−(12+1)=−13
A
13
=(−1)
1+3
∣
∣
∣
∣
∣
∣
2
−1
−3
4
∣
∣
∣
∣
∣
∣
=8−3=5
A
21
=(−1)
2+1
∣
∣
∣
∣
∣
∣
5
4
−4
6
∣
∣
∣
∣
∣
∣
=−(30+16)=−46
A
22
=(−1)
2+2
∣
∣
∣
∣
∣
∣
3
−1
−4
6
∣
∣
∣
∣
∣
∣
=18−4=14
A
23
=(−1)
2+3
∣
∣
∣
∣
∣
∣
3
−1
5
4
∣
∣
∣
∣
∣
∣
=−(12+5)=−17
A
31
=(−1)
3+1
∣
∣
∣
∣
∣
∣
5
−3
−4
1
∣
∣
∣
∣
∣
∣
=5−12=−7
A
32
=(−1)
3+2
∣
∣
∣
∣
∣
∣
3
2
−4
1
∣
∣
∣
∣
∣
∣
=−(3+8)=−11
A
33
=(−1)
3+3
∣
∣
∣
∣
∣
∣
3
2
5
−3
∣
∣
∣
∣
∣
∣
−9−10=−19
AdjA=
⎣
⎢
⎢
⎡
−22
−13
5
−46
14
−17
−7
−11
−19
⎦
⎥
⎥
⎤
A
−1
=
∣A∣
1
adjA=−
151
1
⎣
⎢
⎢
⎡
−22
−13
5
−46
14
−17
−7
−11
−19
⎦
⎥
⎥
⎤
We have X=A
−1
B
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=−
151
1
⎣
⎢
⎢
⎡
−22
−13
5
−46
14
−17
−7
−11
−19
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
6000
5000
13000
⎦
⎥
⎥
⎤
=−
151
1
⎣
⎢
⎢
⎡
−453000
−151000
−302000
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
3000
1000
2000
⎦
⎥
⎥
⎤
so, Price per unit of comodity P=3000 rupees.
Price per unit of comodity Q=1000 rupees.
Price per unit of comodity R=2000 rupees.