Question

The cost of 4 kg onion,3 kg wheat and 2kg 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 1

Let the cost of onions, wheat, and rice per kg be Rs x, Rs y, and Rs z respectively.
Then, the given situation can be represented by a system of equations as:

4x+3y+2z=60
2x+4y+6z=90
6x+2y+3z=70

This system of equations can be written in the form of AX=B,where

A=$\begin{bmatrix}4&3&2\\\2&4&6\\\6&2&3\end{bmatrix}$, X=$\begin{bmatrix}x\\\y\\\z\end{bmatrix}$ and B=$\begin{bmatrix}60\\\90\\\70\end{bmatrix}$.

|A|=4(12-12)-3(6-36)+2(4-24)=0+90-40=50≠0
Now, A11=0, A12=30, A13=-20
A21=-5, A22=0, A23=10
A31=10, A32=-20, A33=10

∴ adj A=$\begin{bmatrix}0&-5&10\\\30&0&-20\\\\-20&10&10\end{bmatrix}$

∴A-1=$\frac{1}{\mid A \mid}$adj A=\frac{1}{50}$$\begin{bmatrix}0&-5&10\\\30&0&-20\\\\-20&10&10\end{bmatrix}
Now, X=A-1 B

$\Rightarrow$ X=\frac{1}{50}$$\begin{bmatrix}0&-5&10\\\30&0&-20\\\\-20&10&10\end{bmatrix}$$\begin{bmatrix}60\\\90\\\70\end{bmatrix}

$\Rightarrow$ $\begin{bmatrix}x\\\y\\\z\end{bmatrix}$=\frac{1}{50}$$\begin{bmatrix}0-450+700\\\1800+0-1400\\\\-1200+900+700\end{bmatrix}

=\frac{1}{50}$$\begin{bmatrix}250\\\400\\\400\end{bmatrix}

=$\begin{bmatrix}5\\\8\\\8\end{bmatrix}$
∴x=5,y=8 and z=8

Hence,the cost of onions is Rs 5 per kg, the cost of wheat is Rs 8 per kg ,and the cost of rice is Rs 8 per kg.