Matrices Modeling Contexts Practice Test

A regional economy tracks how two industries depend on each other. Matrices organize and manipulate this data: each entry tells how many dollars of input from one sector are needed to produce $1 of output in another. For example, the input-output matrix $$\mathbf{A}=\begin{bmatrix}0.20&0.10\0.30&0.40\end{bmatrix}$$ (rows = input sector, columns = output sector) models Manufacturing (row/column 1) and Energy (row/column 2). If total output is $$\mathbf{x}=\begin{bmatrix}100\50\end{bmatrix}$$ (in millions of dollars), then matrix multiplication $\mathbf{A}\mathbf{x}$ gives the total intermediate demand. Matrix addition/subtraction can compare two years’ input-output tables, and multiplication can also represent transformations or systems of equations in other settings.

Using the matrix model described, what does element $(2,1)$ of $\mathbf{A}$ represent?

Matrices organize information and help solve real problems. A theater sells Adult and Student tickets. Revenue is found by multiplying a price row vector by a sales column vector. If prices are $$\mathbf{p}=\begin{bmatrix}12&8\end{bmatrix}$$ (dollars) and sales are $$\mathbf{s}=\begin{bmatrix}150\200\end{bmatrix},$$ then multiplication combines prices and quantities, while addition/subtraction compares sales across days. Using the matrix model described, what operation gives total revenue?