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Costs and Matrices

The table below represents the amounts of different types of materials to build different styles of houses.
Style\Material Steel Wood Glass Paint Concrete
Ranch 5 20 16 7 17
Cape \: Cod 7 18 9 12 21
Colonial 6 25 8 5 13
We can represent this by the matrix  
\[C=\left( \begin{array}{ccccc} 5 & 20 & 16 & 7 & 17 \\ 7 & 18 & 9 & 12 & 21 \\ 6 & 2 & 8 & 5 & 13 \end{array} \right)\]
.
Suppose we are to build 5 Ranch, 7 Cape Cod and 12 Colonial style houses respectively. We can represent this by the vector  
\[\mathbf{v} = \begin{pmatrix}5\\7\\12\end{pmatrix}\]
.
\[\mathbf{v}^TC =(5, 7, 12)\left( \begin{array}{ccccc} 5 & 20 & 16 & 7 & 17 \\ 7 & 18 & 9 & 12 & 21 \\ 6 & 2 & 8 & 5 & 13 \end{array} \right) = (146, 526, 260, 158, 388)\]
.
\[(146, 526, 260, 158, 388) \begin{pmatrix}15\\8\\5\\1\\10\end{pmatrix}=11,736\]
.

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