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

As there are 3 varieties of pen A, B and C

Meenu purchased 1 pen of each variety which costs her Rs 21

Therefore, A+B+C=21Similarly, 

For Jeevan

4A+3B+2C=60

For Shikha

 6A+2B+3C=70

`[[1,1,1],[4,3,2],[6,2,3]][[A],[B],[C]]=[[21],[60],[70]]`

`where P=[[1,1,1],[4,3,2],[6,2,3]], Q=[[21],[60],[70]]`

P|=1(9−4)−1(12−12)+1(8−18)

=−5≠0

∴P⁻¹ existsX=P⁻¹Q

C₁₁=5        

C₁₂=0          

C₁₃=−10

C₂₁=−1    

C₂₂=−3     

C₂₃=4

C₃₁=−1     

C₃₂=2         

C₃₃=−1

`adj P=[[5,0,-10],[-1,-3,4],[-1,2,-1]]^T=[[5,-1,-1],[0,-3,2],[-10,4,-1]]`

`P^-1=1/-5[[5,-1,-1],[0,-3,2],[-10,4,-1]]`

`X=P^-1 Q`

`=1/-5[[5,-1,-1],[0,-3,2],[-10,4,-1]] [[21],[60],[70]]= 1/-5 [[105-60-70],[0-180+140],[-210+240-70]]`

`=-1/5 [[-25],[-40],[-40]]`

`therefore X=[[5],[8],[8]]`

Therefore, cost of A variety of pens =Rs 5

Cost of B variety of pens =Rs 8

Cost of C variety of pens =Rs 8