# 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)
therefore : 10x+5y+15z = 70000
2x+y+3z = 14000 ........(2)
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:

and $X={A}^{-1}B$
therefore first we will find the inverse of matrix A.
$\left|A\right|=1-6+9-2+4-3\phantom{\rule{0ex}{0ex}}=3$

${A}^{-1}=\frac{adj\left(A\right)}{\left|A\right|}\phantom{\rule{0ex}{0ex}}=\left[\begin{array}{c}-5/3\\ 7/3\\ 1/3\end{array}\begin{array}{c}1/3\\ -2/3\\ 1/3\end{array}\begin{array}{c}2/3\\ -1/3\\ -1/3\end{array}\right]$
${A}^{-1}B=\left[\begin{array}{c}-5/3\\ 7/3\\ 1/3\end{array}\begin{array}{c}1/3\\ -2/3\\ 1/3\end{array}\begin{array}{c}2/3\\ -1/3\\ -1/3\end{array}\right]\left[\begin{array}{c}6000\\ 14000\\ 11000\end{array}\right]\phantom{\rule{0ex}{0ex}}=\left[\begin{array}{c}-10000+\frac{14000}{3}+\frac{22000}{3}\\ 14000-\frac{28000}{3}-\frac{11000}{3}\\ 2000+\frac{14000}{3}-\frac{11000}{3}\end{array}\right]$
${A}^{-1}B=\left[\begin{array}{c}-10000+12000\\ 14000-13000\\ 2000+1000\end{array}\right]=\left[\begin{array}{c}2000\\ 1000\\ 3000\end{array}\right]\phantom{\rule{0ex}{0ex}}X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}2000\\ 1000\\ 3000\end{array}\right]$
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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