Question

A shopkeeper has 3 varieties of pens 'A', 'B' and 'C'. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeevan purchased 4 pens of 'A' variety 3 pens of 'B' variety and 2 pens of 'C' variety for Rs 60. While Shikha purchased 6 pens of 'A' variety, 2 pens of 'B' variety and 3 pens of 'C' variety for Rs 70. Using matrix method, find cost of each variety of pen.

Answer 1

$$\begin{gathered} \mathrm{As}\mathrm{there}\mathrm{are}3\mathrm{varieties}\mathrm{of}\mathrm{pen}A,B\mathrm{and}C \\ \mathrm{Meenu}\mathrm{purchased}1\mathrm{pen}\mathrm{of}\mathrm{each}\mathrm{variety}\mathrm{which}\mathrm{costs}\mathrm{her}\mathrm{Rs}21 \\ \mathrm{Therefore}, \\ A+B+C=21 \\ \mathrm{Similarly}, \\ \mathrm{For}\mathrm{Jeevan} \\ 4A+3B+2C=60 \\ \mathrm{For}\mathrm{Shikha} \\ 6A+2B+3C=70 \\ \left[\begin{array}{ccc}1& 1& 1\\ 4& 3& 2\\ 6& 2& 3\end{array}\right]\left[\begin{array}{c}A\\ B\\ C\end{array}\right]=\left[\begin{array}{c}21\\ 60\\ 70\end{array}\right] \\ \mathrm{where}P=\left[\begin{array}{ccc}1& 1& 1\\ 4& 3& 2\\ 6& 2& 3\end{array}\right],Q=\left[\begin{array}{c}21\\ 60\\ 70\end{array}\right] \\ \left|P\right|=1\left(9-4\right)-1\left(12-12\right)+1\left(8-18\right) \\ =-5\ne 0 \\ \therefore P^{-1}\mathrm{exists} \\ X=P^{-1}Q \\ {C}_{11}=5C_{12}=0C_{13}=-10 \\ {C}_{21}=-1C_{22}=-3C_{23}=4 \\ {C}_{31}=-1C_{32}=2C_{33}=-1 \\ \mathrm{adj}P={\left[\begin{array}{ccc}5& 0& -10\\ -1& -3& 4\\ -1& 2& -1\end{array}\right]}^T=\left[\begin{array}{ccc}5& -1& -1\\ 0& -3& 2\\ -10& 4& -1\end{array}\right] \\ {P}^{-1}=\frac{1}{-5}\left[\begin{array}{ccc}5& -1& -1\\ 0& -3& 2\\ -10& 4& -1\end{array}\right] \\ X=P^{-1}Q \end{gathered}$$

$$\begin{gathered} \frac{1}{-5}\left[\begin{array}{ccc}5& -1& -1\\ 0& -3& 2\\ -10& 4& -1\end{array}\right]\left[\begin{array}{c}21\\ 60\\ 70\end{array}\right] \\ =\frac{1}{-5}\left[\begin{array}{c}105-60-70\\ 0-180+140\\ -210+240-70\end{array}\right] \\ =\frac{1}{-5}\left[\begin{array}{c}-25\\ -40\\ -40\end{array}\right] \\ \therefore X=\left[\begin{array}{c}5\\ 8\\ 8\end{array}\right] \\ \mathrm{Therefore},\mathrm{cost}\mathrm{of}\mathrm{A}\mathrm{variety}\mathrm{of}\mathrm{pens}=\mathrm{Rs}5 \\ \mathrm{Cost}\mathrm{of}\mathrm{B}\mathrm{variety}\mathrm{of}\mathrm{pens}=\mathrm{Rs}8 \\ \mathrm{Cost}\mathrm{of}\mathrm{C}\mathrm{variety}\mathrm{of}\mathrm{pens}=\mathrm{Rs}8 \end{gathered}$$