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Question
The prices of three commodities A, B, and C are ₹ x, y, and z per unit respectively. P purchases 4 units of C and sells 3 units of A and 5 units of B. Q purchases 3 units of B and sells 2 units of A and 1 unit of C. R purchases 1 unit of A and sells 4 units of B and 6 units of C. In the process P, Q and R earn ₹ 6,000, ₹ 5,000 and ₹ 13,000 respectively. By using the matrix inversion method, find the prices per unit of A, B, and C.
Solution
Take selling the units js positive earning and buying the units is negative earning.
Given that
3x + 5y – 4z = 6000
2x – 3y + z = 5000
-1x + 4y + 6z = 13000
| A (x) | B (y) | C (z) | |
| P | 3 | 5 | -4 |
| Q | 2 | -3 | 1 |
| R | -1 | 4 | 6 |
The given statement can be written as
`((3,5,-4),(2,-3,1),(-1,4,6)) ((x),(y),(z)) = ((6000),(5000),(13000))`
AX = B
Where A = `((3,5,-4),(2,-3,1),(-1,4,6))`, X = `((x),(y),(z))` and B = `((6000),(5000),(13000))`
X = A-1B
|A| = `|(3,5,-4),(2,-3,1),(-1,4,6)|`
= 3(-18 – 4) – 5(12 + 1) – 4(8 – 3)
= 3(-22) – 5(13) – 4(5)
= -66 – 65 – 20
= -151
`("A"_"ij") = [(-22,-13,5),(-(30+16),18-4,-(12+5)),((5 - 12),-(3 + 8),(-9-10))]`
`= [(-22,-13,5),(-46,14,-17),(-7,-11,-19)]`
adj A = `[(-22,-46,-7),(-13,14,-11),(5,-17,-19)]`
`"A"^-1 = 1/|"A"|` (adj A)
`= 1/(-151) [(-22,-46,-7),(-13,14,-11),(5,-17,-19)]`
X = A-1B
`= 1/(-151) [(-22,-46,-7),(-13,14,-11),(5,-17,-19)] [(6000),(5000),(13000)]`
`= (-1000)/151 [(-22,-46,-7),(-13,14,-11),(5,-17,-19)] [(6),(5),(13)]`
X = `(- 1000)/151 [(-132-230-91),(-78+70-143),(30-85-247)]`
`= (-1000)/151 [(-453),(-151),(-302)]`
=`- 1000 [(-3),(-1),(-2)]`
`[(x),(y),(z)] = [(3000),(1000),(2000)]`
The prices per unit of A, B and C are ₹ 3000, ₹ 1000 and ₹ 2000.
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