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The cost of 4kg onion, 3kg wheat and 2kg rice is Rs. 60. The cost of 2kg onion, 4kg wheat and 6kg rice is Rs. 90. The cost of 6kg onion 2kg wheat and 3kg rice is Rs. 70. Find the cost of each item per kg by matrix method.
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The cost of 4kg onion, 3kg wheat and 2kg rice is Rs. 60. The cost of 2kg onion, 4kg wheat and 6kg rice is Rs. 90. The cost of 6kg onion 2kg wheat and 3kg rice is Rs. 70. Find the cost of each item per kg by matrix method.

Answer

Let  cost of 1kg onion = x
cost of 1kg wheat = y
cost of 1kg rise = z
By the question ,we have,
4x + 3y + 2z = 60
2x + 4y + 6z = 90
6x + 2y + 3z = 70
\(A = \left[ {\begin{array}{*{20}{c}} 4&3&2 \\ 2&4&6 \\ 6&2&3 \end{array}} \right]B = \left[ {\begin{array}{*{20}{c}} {60} \\ {90} \\ {70} \end{array}} \right]X = \left[ {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right]\)
\(\left| A \right| = \left|{\begin{array}{*{20}{c}} 4&3&2 \\ 2&4&6 \\ 6&2&3 \end{array}} \right|= 50 \ne 0\)
\(Now,A_{11}=0,A_{12}=30,A_{13}=-20\)
\(A_{21}=-5,A_{22}=0,A_{23}=10\)
\(A_{31}=10,A_{32}=-20,A_{33}=10\)
\(\therefore adjA = \left[ {\begin{array}{*{20}{c}} 0&{ - 5}&{10} \\ {30}&0&{ - 20} \\ { - 20}&{10}&{10} \end{array}} \right]\)
\({A^{ - 1}} = \frac{1}{{\left| A \right|}}(adjA) = \frac{1}{{50}}\left[ {\begin{array}{*{20}{c}} 0&{ - 5}&{10} \\ {30}&0&{ - 20} \\ { - 20}&{10}&{10} \end{array}} \right]\)
\(X = {A^{ - 1}}B\)
\(\left[ {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 5 \\ 8 \\ 8 \end{array}} \right]\)
x = 5, y = 8, z = 8
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