The prices of three commodities P, Q and R are Rs x, y and z per unit respectively.
Question:

The prices of three commodities PQ and R are Rs xy and z per unit respectively. A purchases 4 units of R and sells 3 units of P and 5 units of QB purchases 3 units of Q and sells 2 units of P and 1 unit of RC purchases 1 unit of P and sells 4 units of Q and 6 units of R. In the process AB and C earn Rs 6000, Rs 5000 and Rs 13000 respectively. If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using matrix method.c

Solution:

The prices of three commodities $P, Q$ and $R$ are Rs $x$, Rs $y$ and Rs $z$ per unit, respectively.

According to the question,

$3 x+5 y-4 z=6000$

$2 x-3 y+z=5000$

$-x+4 y+6 z=13000$

The given system of equations can be written in matrix form as follows:

$\left[\begin{array}{ccc}3 & 5 & -4 \\ 2 & -3 & 1 \\ -1 & 4 & 6\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}6000 \\ 5000 \\ 13000\end{array}\right]$

$A X=B$

Here,

$A=\left[\begin{array}{ccc}3 & 5 & -4 \\ 2 & -3 & 1 \\ -1 & 4 & 6\end{array}\right] X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right] B=\left[\begin{array}{c}6000 \\ 5000 \\ 13000\end{array}\right]$

Now,

$|A|=3(-18-4)-5(12+1)-4(8-3)$

$=-66-65-20$

$=-151 \neq 0$

So, $A^{-1}$ exists.

Let $\mathrm{C}_{i j}$ be the cofactors of the elements $\mathrm{a}_{i j}$ in $\mathrm{A}=\left[a_{i j}\right]$. Then,

$C_{11}=(-1)^{1+1}\left|\begin{array}{cc}-3 & 1 \\ 4 & 6\end{array}\right|=-22, C_{12}=(-1)^{1+2}\left|\begin{array}{cc}2 & 1 \\ -1 & 6\end{array}\right|=-13, C_{13}=(-1)^{1+3}\left|\begin{array}{cc}2 & -3 \\ -1 & 4\end{array}\right|=5$

$C_{21}=(-1)^{2+1}\left|\begin{array}{cc}5 & -4 \\ 4 & 6\end{array}\right|=-46, C_{22}=(-1)^{2+2}\left|\begin{array}{cc}3 & -4 \\ -1 & 6\end{array}\right|=14, C_{23}=(-1)^{2+3}\left|\begin{array}{cc}3 & 5 \\ -1 & 4\end{array}\right|=-17$

$C_{31}=(-1)^{3+1}\left|\begin{array}{cc}5 & -4 \\ -3 & 1\end{array}\right|=-7, C_{32}=(-1)^{3+2}\left|\begin{array}{cc}3 & -4 \\ 2 & 1\end{array}\right|=-11, C_{33}=(-1)^{3+3}\left|\begin{array}{cc}3 & 5 \\ 2 & -3\end{array}\right|=-19$

adj $A=\left[\begin{array}{ccc}-22 & -13 & 5 \\ -46 & 14 & -17 \\ -7 & -11 & -19\end{array}\right]^{T}$

$=\left[\begin{array}{ccc}-22 & -46 & -7 \\ -13 & 14 & -11 \\ 5 & -17 & -19\end{array}\right]$

$\Rightarrow A^{-1}=\frac{1}{|A|} \operatorname{adj} A$

$=\frac{1}{-151}\left[\begin{array}{ccc}-22 & -46 & -7 \\ -13 & 14 & -11 \\ 5 & -17 & -19\end{array}\right]$

$X=A^{-1} B$

$\Rightarrow X=\frac{1}{{ }_{-151}}\left[\begin{array}{ccc}-22 & -46 & -7 \\ -13 & 14 & -11 \\ 5 & -17 & -19\end{array}\right]\left[\begin{array}{c}6000 \\ 5000 \\ 13000\end{array}\right]$

$\Rightarrow X=\frac{1}{-151}\left[\begin{array}{l}-453000 \\ -151000 \\ -302000\end{array}\right]$

$\Rightarrow\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}3000 \\ 1000 \\ 2000\end{array}\right]$

$\therefore x=3000, y=1000$ and $z=2000$

Thus, the prices of the three commodities $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ are Rs 3000, Rs 1000 and Rs 2000 per unit, respectively.