The prices of three commodities P,Q,and R are rupees x,y and z unit respectively.A purchases 4 units of R and sells 3 units of P and 5 units of Q. B purchases 3 units of Q and sells 2 units of P and 1 unit of R, C purchases 1 unit of P and sells 4 units of Q and 6 units of R. In the process A,B,C earns Rs. 6000, 5000 and 13000 respectively. If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using matrix method.

For A: 3x+5y-4z=6000For B: 2x-3y+z=5000For C: -x+4y+6z=13000These equations can be written as AX=B.Where, A=35-42-31-146  , X=xyz  and  B=6000500013000A=3-18-4-512+1-48-3=-66-65-20=-1510Therefore A is nonsingular matrix and inverse exists.A11=-22   A12=-13   A13=5A21=-46   A22=14   A23=-17A31=-7   A32=-11   A33=-19So, adj A=-22-46-7-1314-115-17-19A-1=1Aadj A=-1151-22-46-7-1314-115-17-19X=A-1Bxyz=-1151-22-46-7-1314-115-17-196000500013000       =-1151-453000-151000-302000=300010002000so, Price per unit of comodity P=3000 rupees.Price per unit of comodity Q=1000 rupees.Price per unit of comodity R=2000 rupees.

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