A large apartment building is to be built using modular construction techiniques. The arrangement of apartment on any particular floor is to be chosen from one of three basic floor plans. Plan A has 18 apartments on one floor, including 3 three bedroom units, and 8 one bedroom units. 7 two bedroom units and 8 one bedroom units. Each floor of plan B includes 4 three bedroom units, 4 two bedroom units, and 8 one bedroom units. Each floor of plan C includes 5 three bedroom units, 3 two bedroom units, and 9 one bedroom units. Suppose the building contains a total of \({x_{\bf{1}}}\) floors of plan A, \({x_2}\) floor of plans B, and \({x_{\bf{3}}}\) floors of plan C.

a. What interpretation can be given to the vector \({x_{\bf{1}}}\left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{7}}\\{\bf{8}}\end{aligned}} \right)\)?

b. Write a formal linear combination of vectors that expresses the total numbers of three-, two-, and one-bedroom apartments contained in the building.

c. (M) Is it possible to design the building with exactly 66 three bedroom units, 74 two bedrooms units, and 136 one bedroom units? If so, is there more than one way to do it? Explain your answer.

The vector \({x_1}\left( {\begin{aligned}{*{20}{c}}3\\7\\8\end{aligned}} \right)\) represents the number of three-, two-, and one- bedroom apartments on floor \({x_1}\).

For floors \({x_1}\), \({x_2}\), and \({x_3}\), the linear combination of vectors is

\({x_1}\left( {\begin{aligned}{*{20}{c}}3\\7\\8\end{aligned}} \right) + {x_2}\left( {\begin{aligned}{*{20}{c}}4\\4\\8\end{aligned}} \right) + {x_3}\left( {\begin{aligned}{*{20}{c}}5\\3\\9\end{aligned}} \right)\).

By the linear combination of vectors, you get

\({x_1}\left( {\begin{aligned}{*{20}{c}}3\\7\\8\end{aligned}} \right) + {x_2}\left( {\begin{aligned}{*{20}{c}}4\\4\\8\end{aligned}} \right) + {x_3}\left( {\begin{aligned}{*{20}{c}}5\\3\\9\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{66}\\{74}\\{136}\end{aligned}} \right)\).

Using the equation for the linear combination of vectors, the augmented matrix is

\(\left( {\begin{aligned}{*{20}{c}}3&4&5&{66}\\7&4&3&{74}\\8&8&9&{136}\end{aligned}} \right)\).

Use the code in the MATLAB to obtain the row-reduced echelon form of the augmented matrix \(\left( {\begin{aligned}{*{20}{c}}3&4&5&{66}\\7&4&3&{74}\\8&8&9&{136}\end{aligned}} \right)\).

\(\begin{aligned}{l} > > {\rm{ A }} = {\rm{ }}\left( {{\rm{3 4 5 66}};{\rm{ 7 4 3 74}};{\rm{ 8 8 9 136}};} \right);\\ > > {\rm{ U}} = {\rm{rref}}\left( {\rm{A}} \right)\end{aligned}\)

\(\left( {\begin{aligned}{*{20}{c}}1&0&{ - \frac{1}{2}}&{ - 2}\\0&1&{\frac{{13}}{8}}&{ - 15}\\0&0&0&0\end{aligned}} \right)\)

Then, \({x_1} - \frac{1}{2}{x_3} = 2\) and \({x_2} + \frac{{13}}{8}{x_3} = 15\).

If \({x_3} = 0\), then there are two floors of plan A and 15 floors of plan B.

If \({x_3} = 8\), then there are six floors of plan A, two floors of plan B, and eight floors of plan C.

These are the only feasible solutions. For larger multiples of 8, the number of floors will be negative.

Hence, \({x_1}\) represents the number of different bedrooms. The linear combination of floor vectors is \({x_1}\left( {\begin{aligned}{*{20}{c}}3\\7\\8\end{aligned}} \right) + {x_2}\left( {\begin{aligned}{*{20}{c}}4\\4\\8\end{aligned}} \right) + {x_3}\left( {\begin{aligned}{*{20}{c}}5\\3\\9\end{aligned}} \right)\), and the possible number of floors are:

1. Two floors of plan A, 15 floors of plan B, and zero floors of plan C
2. Six floors of plan A, two floors of plan B, and eight floors of plan C