Case-Study 2: Read the following passage and answer the questions given below

Three friends Ravi, Raju and Rohit were doing buying and selling of stationery items in a market. The price of per dozens of pens, notebooks and toys are Rupees x, y and z respectively. Ravi purchases 4 dozen of notebooks and sells 2 dozen of pens 5 dozen of toys. Raju purchases 2 dozen of toy and sells 3 dozen of pens and 1 dozen of notebooks. Rohit purchases one dozen of pens and sells 3 dozen of notebooks and one dozen of toys.

In the process, Ravi, Raju and Rohit earn Rs. 1500, Rs. 100 and Rs. 400 respectively.

(i) Write the above information in terms of matrix Algebra.

(ii) What is the total price of one dozen of pens and one dozen of notebooks?

(iii) What is the sale amount of Ravi?

                          OR

 What is the amount of purchases and sales made by all three friends?

(i) From the above information,

$\begin{array}{rr} & 2\mathrm{x}-4\mathrm{y}+5\mathrm{z}=1500\\ & 3\mathrm{x}+\mathrm{y}-2\mathrm{z}=100\\ & \mathrm{ }-\mathrm{x}+3\mathrm{y}+\mathrm{z}=400\\ & \left[\begin{array}{ccc}2& -4& 5\\ 3& 1& -2\\ -1& 3& 1\end{array}\right]\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right]=\left[\begin{array}{c}1500\\ 100\\ 400\end{array}\right]\end{array}$

(ii) From the above matrix assume that,

$\begin{array}{rr} & \mathrm{A}=\left[\begin{array}{ccc}2& -4& 5\\ 3& 1& -2\\ -1& 3& 1\end{array}\right],\mathrm{X}=\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right],\mathrm{B}=\left[\begin{array}{c}1500\\ 100\\ 400\end{array}\right]\\ & \mathrm{X}={\mathrm{A}}^{-1}\mathrm{B}\dots \left(\mathrm{i}\right)\\ & \mathrm{ }\left|\mathrm{A}\right|=2(1+6)+4(3-2)+5(9+1)=68\ne 0\\ & \text{ co-factor matrix }\mathrm{A}=\left[\begin{array}{ccc}7& -1& 10\\ 19& 7& -2\\ 3& 19& 14\end{array}\right]\text{, adj }\mathrm{A}=\left[\begin{array}{ccc}7& 19& 3\\ -1& 7& 19\\ 10& -2& 14\end{array}\right]\\ & {\mathrm{A}}^{-1}=\frac{1}{\left|\mathrm{A}\right|}\cdot \text{ adj A}\\ & {\mathrm{A}}^{-1}=\frac{1}{68}\left[\begin{array}{ccc}7& 19& 3\\ -1& 7& 19\\ 10& -2& 14\end{array}\right]\end{array}$From (i)

$\Rightarrow \left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\frac{1}{68}\left[\begin{array}{ccc}7& 19& 3\\ -1& 7& 19\\ 10& -2& 14\end{array}\right]\left[\begin{array}{c}1500\\ 100\\ 400\end{array}\right]$

$\Rightarrow \left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\frac{1}{68}\left[\begin{array}{c}10500+1900+1200\\ -1500+700+7600\\ 15000-200+5600\end{array}\right]$

$\Rightarrow \left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\frac{1}{68}\left[\begin{array}{c}13600\\ 6800\\ 20400\end{array}\right]$

$\mathrm{x}=200,\mathrm{y}=100 \mathrm{and} \mathrm{z}=300$

So, the total price of one dozen of pens and one dozen of notebooks = Rs. 200 + Rs. 100 = Rs. 300.

(iii) For,

The sale amount of Ravi $=2x+5z=2\times 200+5\times 300=400+1500=\mathrm{₹}1900$or 

Now,

The amount of purchases made by all three friends

$4\mathrm{y}+2\mathrm{z}+\mathrm{x}=4\times 100+2\times 300+200=\mathrm{₹}1200$The amount of sales made by all three friends

$2x+5z+3x+y+3y+z=5x+4y+6z=5\times 200+4\times 100+6\times 300=\mathrm{₹}3200$Therefore, (i) The given information in terms of matrix Algebra it follows,

$\left[\begin{array}{ccc}2& -4& 5\\ 3& 1& -2\\ -1& 3& 1\end{array}\right]\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1500\\ 100\\ 400\end{array}\right]$

(ii) The total price of one dozen of pens and one dozen of notebooks= Rs. 300 .

(iii) The sale amount of Ravi= Rs. 1900 OR the amount of purchases and sales made by all three friends = Rs. 3200 .