Question

The cost of $4 \,kg$ onion, $3\, kg$ wheat and $2 \,kg$ rice is ₹ 60 . The cost of $2 \,kg$ onion, $4\, kg$ wheat and $6 \,kg$ rice is ₹ 90 . The cost of $6 kg$ onion, $2\, kg$ wheat and $3\, kg$ rice is $₹ 70$. The costs (in $₹$ ) of onion, wheat and rice per kilogram are respectively

• A. $5,9,10$
• B. $5,8,8$
• C. $8,5,8$
• D. $8,8,5$

Answer 1

Answer: (B)

Let the prices (per $kg$ ) of onion, wheat and rice be $₹ x$, $₹ y$ and $₹ z$, respectively. Then,
$ 4 x+3 y+2 z=60 $
$2 x+4 y+6 z=90$
$ 6 x+2 y+3 z=70$
This system of equations can be written as $A X=B$, where
$A = \begin{bmatrix}4 & 3 & 2 \\2 & 4 & 6 \\6 & 2 & 3\end{bmatrix}, X=\begin{bmatrix}x \\y \\z\end{bmatrix} \text { and } B=\begin{bmatrix}60 \\90 \\70\end{bmatrix} $
Here, $|A| = \begin{vmatrix} 4 & 3 & 2 \\2 & 4 & 6 \\6 & 2 & 3\end{vmatrix} $
$=4(12-12)-3(6-36)+2(4-24).$
$=0+90-40=50 \neq 0$
Thus, $A$ is non-singular. Therefore, its inverse exists. Therefore, the given system is consistent and has a unique solution given by $X=A^{-1} B$.
Cofactors of $A$ are
$ A_{11}=12-12=0, A_{12}=-(6-36)=30, A_{13}=4-24=-20,$
$A_{21}=-(9-4)=-5, A_{22}=12-12=0, A_{23}=-(8-18)=10,$
$A_{31}=(18-8)=10, A_{32}=-(24-4)=-20, A_{33}=16-6=10 $
$\operatorname{adj}(A)=\begin{bmatrix}0 & 30 & -20 \\-5 & 0 & 10 \\10 & -20 & 10\end{bmatrix}^{\prime}=\begin{bmatrix}0 & -5 & 10 \\30 & 0 & -20 \\-20 & 10 & 10\end{bmatrix} $
$\therefore A^{-1}=\frac{1}{|A|}(\operatorname{adj}(A)=\frac{1}{50}\begin{bmatrix}0 & -5 & 10 \\30 & 0 & -20 \\-20 & 10 & 10\end{bmatrix} $
Now,$ X=A^{-1} B \Rightarrow\begin{bmatrix}x \\y \\z\end{bmatrix}=\frac{1}{50}\begin{bmatrix}30 & -5 & 10 \\-20 & 10 & -20 \\0-450+700\end{bmatrix}\begin{bmatrix}60 \\90 \\70\end{bmatrix} $
$= \frac{1}{50} \begin{bmatrix} 0 - 450 +700\\ 1800+0-1400 \\-1200+900+700\end{bmatrix}$
$ \Rightarrow \begin{bmatrix}x \\ y \\ z \end{bmatrix}=\frac{1}{50}\begin{bmatrix}250 \\400 \\400\end{bmatrix}=\begin{bmatrix}5 \\8 \\8\end{bmatrix} $
$\Rightarrow \begin{bmatrix} x \\y \\z\end{bmatrix}=\begin{bmatrix}5 \\ 8 \\8\end{bmatrix} $
$ \therefore x=5, y=8 \text { and } z=8 .$
Hence, price of onion per $kg$ is $₹ 5$, price of wheat per $kg$ is ₹ 8 and price of rice per $kg$ is $₹ 8$.