Problem 52 - Algebra and Trigonometry

Question

Produce Sales A farmer's three children, Amy, Beth, and Chad, run three roadside produce stands during the summer months. One weekend they all sell watermelons, yellow squash, and tomatoes. The matrices \(A\) and \(B\) tabulate the number of pounds of each product sold by each sibling on Saturday and Sunday. The matrix \(C\) gives the price per pound (in dollars) for each type of produce that they sell. Perform the following matrix operations, and interpret the entries in each result. \(\begin{array}{llll}{\text { (a) } A C} & {\text { (b) } B C} & {\text { (c) } A+B} & {\text { (d) }(A+B) C}\end{array}\)

Step-by-Step Solution

Answer

Calculate total revenue for each child per day and over the weekend.

Step 1: Understand the Problem

We have three matrices: \(A\), which represents the amount of each product (watermelons, yellow squash, and tomatoes) sold by each child on Saturday; \(B\), which represents the amounts sold on Sunday; and \(C\), which contains the price per pound of each product. Our task is to perform matrix operations \(AC\), \(BC\), \(A+B\), and \((A+B)C\).

Step 2: Define the Matrices

Let's assume matrix \(A\) is defined as:\[A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix}\]where rows represent Amy, Beth, and Chad, and columns represent watermelons, yellow squash, and tomatoes.Matrix \(B\) is:\[B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \ b_{21} & b_{22} & b_{23} \ b_{31} & b_{32} & b_{33} \end{bmatrix}\]Matrix \(C\) for price per pound could be:\[C = \begin{bmatrix} c_1 \ c_2 \ c_3\end{bmatrix}\]

Step 3: Perform Matrix Multiplication (a) AC

To find \(AC\), multiply matrix \(A\) with \(C\):\[AC = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix} \cdot \begin{bmatrix} c_1 \ c_2 \ c_3\end{bmatrix} = \begin{bmatrix} a_{11}c_1 + a_{12}c_2 + a_{13}c_3 \ a_{21}c_1 + a_{22}c_2 + a_{23}c_3 \ a_{31}c_1 + a_{32}c_2 + a_{33}c_3\end{bmatrix}\]The resulting matrix indicates the total revenue each person (Amy, Beth, Chad) earns on Saturday from selling all products.

Step 4: Perform Matrix Multiplication (b) BC

Similarly, matrix \(BC\) is obtained by multiplying \(B\) with \(C\):\[BC = \begin{bmatrix} b_{11} & b_{12} & b_{13} \ b_{21} & b_{22} & b_{23} \ b_{31} & b_{32} & b_{33} \end{bmatrix} \cdot \begin{bmatrix} c_1 \ c_2 \ c_3\end{bmatrix} = \begin{bmatrix} b_{11}c_1 + b_{12}c_2 + b_{13}c_3 \ b_{21}c_1 + b_{22}c_2 + b_{23}c_3 \ b_{31}c_1 + b_{32}c_2 + b_{33}c_3\end{bmatrix}\]This result gives the total revenue each child earns on Sunday.

Step 5: Perform Matrix Addition (c) A+B

For \(A+B\), add matrices \(A\) and \(B\):\[A+B = \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} & a_{13}+b_{13} \ a_{21}+b_{21} & a_{22}+b_{22} & a_{23}+b_{23} \ a_{31}+b_{31} & a_{32}+b_{32} & a_{33}+b_{33} \end{bmatrix}\]This matrix represents the total amount (in pounds) of each product sold by each child over the weekend.

Step 6: Perform Matrix Multiplication (d) ((A+B)C)

Multiply the result from (c) with matrix \(C\):\[(A+B)C = \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} & a_{13}+b_{13} \ a_{21}+b_{21} & a_{22}+b_{22} & a_{23}+b_{23} \ a_{31}+b_{31} & a_{32}+b_{32} & a_{33}+b_{33} \end{bmatrix} \cdot \begin{bmatrix} c_1 \ c_2 \ c_3\end{bmatrix} = \begin{bmatrix} (a_{11}+b_{11})c_1 + (a_{12}+b_{12})c_2 + (a_{13}+b_{13})c_3 \ (a_{21}+b_{21})c_1 + (a_{22}+b_{22})c_2 + (a_{23}+b_{23})c_3 \ (a_{31}+b_{31})c_1 + (a_{32}+b_{32})c_2 + (a_{33}+b_{33})c_3\end{bmatrix}\]This matrix shows the combined total revenue for both days for each child.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation used in many areas of mathematics, including statistics, physics, and economics. It involves combining two matrices to produce a new matrix. This process is not as straightforward as regular multiplication. You need to follow a specific rule for combining the rows and columns of the given matrices.

Here's how it works when multiplying matrix \(A\) with matrix \(C\):

• Matrix \(A\) contains rows representing individuals (Amy, Beth, Chad) and columns for products (watermelons, yellow squash, tomatoes).
• Matrix \(C\) is a single row matrix (also known as a vector) representing the prices per pound of each product.
• To multiply \(A\) by \(C\), for each person (row in \(A\)), multiply each amount by the corresponding price in \(C\), and sum these products to get a single number. This number represents the total revenue for that person.

For example, if Amy sold \(m\) pounds of watermelons, \(n\) pounds of yellow squash, and \(p\) pounds of tomatoes, you multiply and sum: \(m \, c_1 + n \, c_2 + p \, c_3\). This process is repeated for each row of matrix \(A\) to find the revenue for each individual.

Matrix Addition

Matrix addition is a simpler matrix operation compared to multiplication. It involves adding corresponding elements of two matrices of the same size to produce a new matrix. In this context, the matrices \(A\) and \(B\) represent the quantities of different products sold on Saturday and Sunday, respectively.

Here's how you perform matrix addition:

• Verify that matrices are of the same dimensions (they both must have the same number of rows and columns).
• Take two matrices, \(A\) and \(B\), and add their corresponding elements (e.g., the amount sold by Amy for a particular product on both days).
• The result is a new matrix where each element represents the total amount of a product sold over the weekend by each child.

This operation allows us to summarize data and gather more comprehensive insights. For example, \(a_{11} + b_{11}\) would give the total amount of watermelons sold by Amy over the entire weekend.

Revenue Calculation

Revenue calculation using matrices is an effective way to organize and compute total incomes from various sales activities, especially when dealing with multiple sets of data.

In this exercise, matrices help to calculate revenues by encapsulating the sold quantities and prices within structured arrays.

• First, perform matrix multiplication \(AC\) and \(BC\) separately to determine revenues for Saturday and Sunday.
• Each multiplication results in a matrix representing each person's individual revenue, based on the quantities they sold (from \(A\) or \(B\)) and the fixed prices (from \(C\)).
• Subsequently, sum these revenues for Saturday and Sunday together using another matrix operation \((A+B)C\), after first performing \(A+B\) to get total quantities per product.

By organizing data this way, calculating revenue becomes more systematic, accurate, and easier to interpret. This matrix approach finds a broad application in financial analysis and business, where understanding the financial outcome, based on sales data, is crucial.