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.

Answer 1

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.