Three shopkeepers A, B, C are using polythene, handmade bags (prepared by prisoners), andnewspapers envelope as carry bags it - Maths - Matrices

Question

Three shopkeepers A, B, C are using polythene, handmade bags
(prepared by prisoners), andnewspapers envelope as carry bags it is
found that the shopkeepers A, B, C are using(20,30,40) ,
(30,40,20,) , (40,20,30) polythene , handmade bags and newspapers
envelopesrespectively The shopkeepers A, B, C spent Rs 250, Rs 220
Rs 200 on these carry bagsrespectively Find the cost of each carry
bags using matrices Keeping in mind the social environmental
conditions, which shopkeeper is better? why?

Utsav · Accepted Answer

Let the cost of polythene ,handmade bags and newspapers envelope be
x,y and z respectively∴20x+30y+40z=250 (i)30x+40y+20z=220
(ii)40x+20y+30z=200 (iii)Let A =203040304020402030, X =xyz and B =
250220200Then the given system in matrix form is AX=BNow
A=20(800)-30(100)+40(-1000)=-27000 =≠0Hence A is invertibleSo
the given system has a unique solution X=A-1BThe cofactors of the
elements of AareM11=800 ,M12=-100 ,M13= -1000 M21=-100 M22=-1000
,M23=800M31=-1000 ,M32=800 ,M33= -100∴(adjA)=
800-100-1000-100-1000800-1000800-100T=800-100-1000-100-1000800-1000800-100⇒A-1=1A
(adjA)=-127000800-100-1000-100-1000800-1000800-100⇒X=A-1B⇒xyz=-12708-1-10-1-108-108-1
250220200 =-1270-220-850-940=0 8143 1483 481Hence x=0 814,y=3
148,z=3 481

∴Cost of polythene,handmade bags and newspaper envelope are
Rs 0 814,Rs 3 148 and Rs 3 481 respectively