Problem 56 - Precalculus: Mathematics for Calculus

Question

A small fast-food chain with restaurants in Santa Monica, Long Beach, and Anaheim sells only hamburgers, hot dogs, and milk shakes. On a certain day, sales were distributed according to the following matrix. $$\begin{array}{rccc} & \text { Number of items sold } \\ \hline \text { Santa } & \text { Long } & \\ \text { Monica } & \text { Beach } & \text { Anaheim }\end{array}$$ $$\begin{aligned}&\begin{array}{c}\text { Hamburgers } \\ \text { Hot dogs } \\\\\ \text { Milk shakes } \end{array}\left[\begin{array}{r} 4000 & 1000 & 3500 \\ 400 & 300 & 200 \\ 700 & 500 & 9000 \end{array}\right]=A\end{aligned}$$ $$ \text { Hamburger } \quad \text { Hot dog } \qquad \text { Milk shake}$$ $$\left[\begin{array}{llll}\$ 0.90 &&&& \$ 0.80 &&& \$ 1.10\end{array}\right]=B$$ (a) Calculate the product \(BA\). (b) Interpret the entries in the product matrix \(BA\).

Step-by-Step Solution

Answer

Matrix \( BA \) represents total revenues: Santa Monica $4690, Long Beach $1690, Anaheim $13210.

Step 1: Set Up the Problem

We are given two matrices, Matrix \( A \) which represents the number of items sold in each location and Matrix \( B \) which represents the prices of each item. We need to calculate the product of matrices \( B \) and \( A \), \( BA \).

Step 2: Determine Matrix Dimensions

Matrix \( A \) is a \( 3 \times 3 \) matrix and Matrix \( B \) is a \( 1 \times 3 \) matrix. The resulting matrix \( BA \) will have dimensions \( 1 \times 3 \) because the number of rows in \( B \) matches the number of columns in \( A \).

Step 3: Calculate Matrix Product

We will multiply each entry in the row of \( B \) by the corresponding entry in each column of \( A \), and sum the results:- First column of \( BA \): \[ 0.90 \times 4000 + 0.80 \times 400 + 1.10 \times 700 = 3600 + 320 + 770 = 4690 \]- Second column of \( BA \): \[ 0.90 \times 1000 + 0.80 \times 300 + 1.10 \times 500 = 900 + 240 + 550 = 1690 \]- Third column of \( BA \): \[ 0.90 \times 3500 + 0.80 \times 200 + 1.10 \times 9000 = 3150 + 160 + 9900 = 13210 \]

Step 4: Write the Resulting Matrix

The product matrix \( BA \) is:\[\left[\begin{array}{ccc}4690 & 1690 & 13210\end{array}\right]\]

Step 5: Interpret the Results

Each entry in the product matrix \( BA \) represents the total revenue generated from each location. - The first entry, 4690, is the revenue from Santa Monica.- The second entry, 1690, is the revenue from Long Beach.- The third entry, 13210, is the revenue from Anaheim.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation where two matrices are combined to produce a new matrix. In this operation, every element of the resulting matrix is found by taking the dot product of corresponding rows and columns from the two matrices being multiplied. Here's how it works:

• Each element in the resulting matrix is calculated as the sum of products of corresponding elements from the row of the first matrix and the column of the second.
• To multiply two matrices, say Matrix \(A\) with dimensions \(m \times n\) and Matrix \(B\) with dimensions \(n \times p\), you need that the number of columns in \(A\) matches the number of rows in \(B\). The resulting matrix will have dimensions \(m \times p\).

In our exercise, we are multiplying matrix \(B\), a \(1 \times 3\) matrix, with matrix \(A\), a \(3 \times 3\) matrix. The multiplication is valid because the number of columns in \(B\) matches the number of rows in \(A\). The result is a \(1 \times 3\) matrix representing revenues from three locations.

Revenue Calculation

Revenue calculation involves determining the total income generated from sales before any expenses are deducted. When employing matrices for revenue calculation, each entry in the resulting matrix represents the revenue from different categories or locations.
Matrix multiplication is a useful tool here because it allows us to multiply a matrix of unit prices with a matrix of quantities sold, resulting in a matrix of revenues.

• For each product and location, we multiply the number of items sold by the price of the item.
• We then sum these products to find the total revenue for each location.

For instance, multiplying the price matrix \(B\) by the quantity matrix \(A\) yielded the revenue matrix \(BA\), where each entry corresponds to the total revenue from each location. This gives us a clear understanding of how each outlet performs financially.

Matrices in Business

Matrices are an invaluable tool in business for organizing and analyzing data efficiently. They allow for the manipulation of large data sets and provide a structured framework to perform complex calculations.
In the context of a business like our fast-food chain, matrices help streamline operations by:

• Consolidating sales data from different locations or categories, as seen in the sales matrix \(A\).
• Facilitating quick and accurate revenue calculations through matrix multiplication, as done when deriving matrix \(BA\).
• Offering a clear visual representation of data, which assists in identifying trends and guiding strategic decisions.

Using matrices allows businesses to efficiently handle different operational aspects such as inventory management, financial analysis, and supply chain logistics. As businesses deal with large quantities of data, matrix operations become essential to maintain competitiveness and operational excellence.