Two Trusts A and B receive Rs 70000 and 55000 respectively from central government to award prize to persons of a district in three fields of agriculture, education, and social service .Trust A awarded 10, 5,and 15 persons in the field of agriculture, education and social service respectively while trust B awarded 15,10,and 5 persons respectively. If all three prizes together amount to 6000, then find the amount of each prize by matrix method ?

let the amount of each prize in the field of agriculture, education and social service be  x, y and z respectively.
the total of three prizes amount to 6000.
therefore: x+y+z = 6000..........(1)
trust A receive Rs. 70000.
therefore : 10x+5y+15z = 70000
2x+y+3z = 14000 ........(2)
trust B receive Rs. 55000/-
therefore : 15x+10y+5z = 55000
3x+2y+z = 11000 ..........(3)
therefore writing the eq(1), eq(2) and eq(3) in the form of matrices:
111213321xyz=60001400011000i.e. AX = Bwhere A=111213321 , X=xyz and B=60001400011000
and X=A-1B 
therefore first we will find the inverse of matrix A.
A=1-6+9-2+4-3=3
cofactor matrix of A=(1-6)-(2-9)(4-3)-(1-2)(1-3)-(2-3)(3-1)-(3-2)(1-2)=-5127-2-111-1
adjoint(A) = transpose of cofactor matrix=-5711-212-1-1
A-1=adj(A)A=-5/37/31/31/3-2/31/32/3-1/3-1/3
A-1B=-5/37/31/31/3-2/31/32/3-1/3-1/360001400011000=-10000+140003+22000314000-280003-1100032000+140003-110003
A-1B=-10000+1200014000-130002000+1000=200010003000X=xyz=200010003000
therefore x = 2000/- y = 1000/- and z = 3000/-
thus the amount of prize in the field of agriculture, education and social service are 2000/-, 1000/- and 3000/-
respectively.

hope this helps you

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