Question

The cost of 2 Kg of Wheat and 1 Kg of Sugar is ₹ 70. The cost of 1 Kg of Wheat and 1 Kg of Rice is ₹ 70. The cost of 3 Kg of Wheat, 2 Kg of Sugar and 1 Kg of rice is ₹ 170. Find the cost of per kg each item using the matrix inversion method.

Answer 1

Let the cost of 1 kg of wheat, sugar and rice be x, y, z respectively. 

By given data, 

2x + y = 70

x + z = 70

3x + 2y + z = 170

`=> [(2,1,0),(1,0,2),(3,2,1)] [(x),(y),(z)] = [(70),(70),(170)]`

⇒ AX = B

Where A = `[(2,1,0),(1,0,2),(3,2,1)]`, X = `[(x),(y),(z)]`, B = `[(70),(70),(170)]`

⇒ X = A^-1B

`|"A"| = |(2,1,0),(1,0,1),(3,2,1)|`

`= 2|(0,1),(2,1)| - 1|(1,1),(3,1)| + 0|(1,0),(3,2)|`

= 2(0 - 2) - 1(1 - 3) + 0

= - 4 + 2 + 0

= - 2 ≠ 0

∴ `"A"^-1` exists

adj A = `[(+|(0,1),(2,1)| -|(1,1),(3,1)| +|(1,0),(3,2)|),
(-|(1,0),(2,1)| + |(2,0),(3,1)| - |(2,1),(3,2)|),(+|(1,0),(0,1)|-|(2,0),(1,1)| +|(2,1),(1,0)|)]^"T"`

`= [(+(0-2),-(1-3),+(2-0)),(-(1-0),+(2-0),-(4-3)),(+(1-0),-(2-0),+(0-1))]^"T"`

`= [(-2,2,2),(-1,2,-1),(1,-2,-1)]^"T"`

`= [(-2,-1,1),(2,2,-2),(2,-1,-1)]`

`"A"^-1 = 1/|"A"|`adj A

`= (-1)/2[(-2,-1,1),(2,2,-2),(2,-1,-1)]`

X = `"A"^-1"B"`

`[(x),(y),(z)] = (-1)/2[(-2,-1,1),(2,2,-2),(2,-1,-1)] [(70),(70),(170)]`

`[(x),(y),(z)] = (-1)/2 [(-140-70+170),(140+140-340),(140-70-170)]`

`[(x),(y),(z)] = (-1)/2 [(-40),(-60),(-100)]`

`[(x),(y),(z)] = [(20),(30),(50)]`

∴ Cost of 1 kg of wheat is Rs. 20.

Cost of 1 kg of Sugar is Rs. 30.

Cost of 1 kg of Rice is Rs. 50.