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  • 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 f

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.

 

 

 

 
 
 
 
 

Answers (1)

Let the cost of each variety of pen be Rs x, Rs y and Rs z resp. Then, x+y+z= 21
4x+3y+2z= 60 & 6x+2y+3= 70
This system of equation can be written in matrix form as follows:
\begin{bmatrix} 1 & 1&1 \\ 4 & 3 &2 \\ 6 & 2 & 3 \end{bmatrix}\begin{bmatrix} x\\ y \\ z \end{bmatrix}= \begin{bmatrix} 21\\ 60 \\ 70 \end{bmatrix}
or AX = B
where A= \begin{bmatrix} 1 & 1 &1 \\ 4 & 3 &2 \\ 6 & 2 & 3 \end{bmatrix},x= \begin{bmatrix} x\\ y \\ z \end{bmatrix}and\, B= \begin{bmatrix} 21\\ 60 \\ 70 \end{bmatrix}
Now, \left | A \right |= \begin{bmatrix} 1 & 1 &1 \\ 4 & 3 &2 \\ 6 & 2 & 3 \end{bmatrix}= 1\left ( 9-4 \right )-1\left ( 12-12 \right )+1\left ( 8-18 \right )
                                           = -5\neq 0
so A-1 exists and the solution of the given system of equation is given by X =A-1B
Let Cij be the cofactor of aij in A=\left | aij \right |. Then,
C_{11}= 5,C_{12}= 0,C_{13}= -10,C_{21}= -1,C_{22}= -3,C_{23}= 4,C_{31}= -1,C_{32}= 2,C_{33}= -1
\therefore adj\, A=\begin{bmatrix} 5 & 0 &-10 \\ -1& -3 &4 \\ -1& 2 & -1 \end{bmatrix}^{T}= \begin{bmatrix} 5 & -1& -1\\ 0 & -3 & 2\\ -10&4 & -1 \end{bmatrix}
so, A^{-1}= \frac{1}{\left | A \right |}\left ( adj A \right )= \frac{-1}{5} \begin{bmatrix} 5 & -1& -1\\ 0 & -3 & 2\\ -10&4 & -1 \end{bmatrix}
Hence the solution is given by
X= A^{-1}B= \frac{-1}{5} \begin{bmatrix} 5 & -1& -1\\ 0 & -3 & 2\\ -10&4 & -1 \end{bmatrix}\begin{bmatrix} 21\\ 60 \\ 70 \end{bmatrix}
= \frac{-1}{5} \begin{bmatrix} 105 -60 -70\\ 0 -180 +140\\ -210+240 -70 \end{bmatrix}\Rightarrow \frac{-1}{5}\begin{bmatrix} -25\\ -40 \\ -40 \end{bmatrix}= \begin{bmatrix} 5\\ 8 \\ 8 \end{bmatrix}
Hence the cost of each variety of pen are Rs 5, Rs 8 and Rs 8 respectively
 

Posted by

Ravindra Pindel

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