The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion,4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

Question

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion,4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

Filo · Accepted Answer

Let the cost of 1 kg onion , 1 kg wheat and 1 kg of rice be

R s x, R s y, R s z

respectively.

So,

4 x + 3 y + 2 z = 60

2 x + 4 y + 6 z = 90

6 x + 2 y + 3 z = 70

These equations can be written as

A X = B

where

A = ⎣ ⎡ 426342263 ⎦ ⎤, X = ⎣ ⎡ x y z ⎦ ⎤, B = ⎣ ⎡ 609070 ⎦ ⎤

Here,

∣ A ∣ = 4 (12 - 12) - 3 (6 - 36) + 2 (4 - 24)

\Rightarrow ∣ A ∣ = 90 - 40 = 50

Since,

∣ A ∣ \neq = 0

Hence, the system of equations is consistent and has a unique solution given by

X == A^{- 1} B

A^{- 1} = \frac{a d j A}{∣ A ∣}

and

a d j A = C^{T}

C_{11} = (- 1)^{1 + 1} ∣ ∣ 4263 ∣ ∣

\Rightarrow C_{11} = 12 - 12 = 0

C_{12} = (- 1)^{1 + 2} ∣ ∣ 2663 ∣ ∣

\Rightarrow C_{12} = - (6 - 36) = 30

C_{13} = (- 1)^{1 + 3} ∣ ∣ 2642 ∣ ∣

\Rightarrow C_{13} = 4 - 24 = - 20

C_{21} = (- 1)^{2 + 1} ∣ ∣ 3223 ∣ ∣

\Rightarrow C_{21} = - (9 - 4) = - 5

C_{22} = (- 1)^{2 + 2} ∣ ∣ 4623 ∣ ∣

\Rightarrow C_{22} = 12 - 12 = 0

C_{23} = (- 1)^{2 + 3} ∣ ∣ 4632 ∣ ∣

\Rightarrow C_{23} = - (8 - 18) = 10

C_{31} = (- 1)^{3 + 1} ∣ ∣ 3426 ∣ ∣

\Rightarrow C_{31} = 18 - 8 = 10

C_{32} = (- 1)^{3 + 2} ∣ ∣ 4226 ∣ ∣

\Rightarrow C_{32} = - (24 - 4) = - 20

C_{33} = (- 1)^{3 + 3} ∣ ∣ 4234 ∣ ∣

\Rightarrow C_{33} = 16 - 6 = 10

Hence, the co-factor matrix is

C = ⎣ ⎡ 0 - 5 10 300 - 20 - 20 1010 ⎦ ⎤

\Rightarrow a d j A = C^{T} = ⎣ ⎡ 030 - 20 - 5 010 10 - 20 10 ⎦ ⎤

\Rightarrow A^{- 1} = \frac{a d j A}{∣ A ∣} = \frac{1}{50} ⎣ ⎡ 030 - 20 - 5 010 10 - 20 10 ⎦ ⎤

Solution is given by

⎣ ⎡ x y z ⎦ ⎤ = \frac{1}{50} ⎣ ⎡ 030 - 20 - 5 010 10 - 20 10 ⎦ ⎤ ⎣ ⎡ 609070 ⎦ ⎤

⎣ ⎡ x y z ⎦ ⎤ = \frac{1}{50} ⎣ ⎡ 0 - 450 + 700 1800 + 0 - 1400 - 1200 + 900 + 700 ⎦ ⎤

⎣ ⎡ x y z ⎦ ⎤ = \frac{1}{50} ⎣ ⎡ 250400400 ⎦ ⎤

⎣ ⎡ x y z ⎦ ⎤ = ⎣ ⎡ 588 ⎦ ⎤

Hence,

x = 5, y = 8, z = 8

So, the cost of

1

kg onion is Rs

5

,

1

kg wheat is Rs

8

,

1

kg rice is Rs

8

.

Question Text	The cost of $4$ kg onion, $3$ kg wheat and $2$ kg rice is Rs $60$ . The cost of $2$ kg onion, $4$ kg wheat and $6$ kg rice is Rs $90$ . The cost of $6$ kg onion $2$ kg wheat and $3$ kg rice is Rs $70$ . Find cost of each item per kg by matrix method.
Answer Type	Text solution:1
Upvotes	150

The cost of $4$ kg onion, $3$ kg wheat and $2$ kg rice is Rs $60$ . The cost of $2$ kg onion, $4$ kg wheat and $6$ kg rice is Rs $90$ . The cost of $6$ kg onion $2$ kg wheat and $3$ kg rice is Rs $70$ . Find cost of each item per kg by matrix method.

Text solutionVerified

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