A contractor builds two types of houses. The costs for labor and materials for the economy model and the deluxe model in thousands of dollars are shown in the tabile. Write the information in the table as a matrix A. Suppose that the contractor built 3 economy models and 7 deluxe models. Write a matrix $Q$ of the appropriate size containing the quanfity of each fype. Find the product matr AQ. What do the entries of $A Q$ represent? \begin{tabular}{|l|c|c|} \hline & Economy & Deluxe \\ \hline Labor Cost & $\$ 21$ & $\$ 40$ \\ Material Cost & $\$ 39$ & $\$ 70$ \\ \hline \end{tabular} Write the matrix $A$ in terms of thousands of dollars. $A=\square($ Simplify your answer.) Write the matrix $Q$. \[ Q=\square \text { (Simplity your answer.) } \] Find the product $A Q$ in terms of thousands of dollars. \[ A Q=\square(\text { Simplity your answer.) } \] What do the entries of $A Q$ represent? Select all that apply. A. The total cost of lahor for 3 economy modols and 7 detuxe models in thousands of dollars B. The total cost of labor and materials for 3 economy models in thousands of dollars C. The total cost of materials for 3 economy models and 7 deluxe models in thousands of dollars D. The total cost of labor and materials for 7 deluxe models in thousands of dollars
Solution
Step 1 ：Define the matrix A as the costs for labor and materials for the economy model and the deluxe model in thousands of dollars. The matrix A is \(\begin{bmatrix} 21 & 40 \ 39 & 70 \end{bmatrix}\).
Step 2 ：Define the matrix Q as the quantity of each type of house built. The matrix Q is \(\begin{bmatrix} 3 \ 7 \end{bmatrix}\).
Step 3 ：Calculate the product of matrices A and Q. The product AQ is \(\begin{bmatrix} 343 \ 607 \end{bmatrix}\).
Step 4 ：The first entry of AQ, 343, represents the total cost of labor for 3 economy models and 7 deluxe models in thousands of dollars.
Step 5 ：The second entry of AQ, 607, represents the total cost of materials for 3 economy models and 7 deluxe models in thousands of dollars.
Step 6 ：Final Answer: Matrix A is \(\boxed{\begin{bmatrix} 21 & 40 \ 39 & 70 \end{bmatrix}}\), Matrix Q is \(\boxed{\begin{bmatrix} 3 \ 7 \end{bmatrix}}\), and Product AQ is \(\boxed{\begin{bmatrix} 343 \ 607 \end{bmatrix}}\). The entries of AQ represent the total cost of labor and materials for 3 economy models and 7 deluxe models in thousands of dollars.
