Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12

# 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. - Mathematics

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.

#### Solution

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(94)1(1212)+1(818)

=50

P1 existsX=P1Q

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

Concept: Inverse of Matrix - Inverse of a Nonsingular Matrix by Elementary Transformation
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