Problem 48 - 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 each of 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 matrix multiplications \( AC \), \( BC \), and \((A+B)C\), and the sum \( A+B \) following the steps above for revenue and produce totals.

Step 1: Define the matrices

Let's assume the matrices as follows: \( A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix} \) represents the pounds of watermelons, yellow squash, and tomatoes sold by Amy, Beth, and Chad on Saturday. Matrix \( B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \ b_{21} & b_{22} & b_{23} \ b_{31} & b_{32} & b_{33} \end{bmatrix} \) represents the same for Sunday. Matrix \( C = \begin{bmatrix} c_1 \ c_2 \ c_3 \end{bmatrix} \) represents the price per pound for each produce.

Step 2: Calculate \( AC \)

Perform the multiplication of matrix \( A \) by matrix \( C \). The result will be a column matrix with entries \( AC = \begin{bmatrix} a_{11}c_1 + a_{12}c_2 + a_{13}c_3 \ a_{21}c_1 + a_{22}c_2 + a_{23}c_3 \ a_{31}c_1 + a_{32}c_2 + a_{33}c_3 \end{bmatrix} \). Each entry represents the total revenue from Amy, Beth, and Chad on Saturday, respectively.

Step 3: Calculate \( BC \)

Perform the multiplication of matrix \( B \) by matrix \( C \). The result will be a column matrix with entries \( BC = \begin{bmatrix} b_{11}c_1 + b_{12}c_2 + b_{13}c_3 \ b_{21}c_1 + b_{22}c_2 + b_{23}c_3 \ b_{31}c_1 + b_{32}c_2 + b_{33}c_3 \end{bmatrix} \). Each entry represents the total revenue from Amy, Beth, and Chad on Sunday, respectively.

Step 4: Calculate \( A + B \)

Add matrices \( A \) and \( B \) to find the total pounds of each type of produce sold by each sibling over the weekend. The result is \( A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & a_{13} + b_{13} \ a_{21} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23} \ a_{31} + b_{31} & a_{32} + b_{32} & a_{33} + b_{33} \end{bmatrix} \).

Step 5: Calculate \((A + B)C\)

Perform the multiplication of the matrix resulting from \( A + B \) with \( C \). The result is a new column matrix: \( (A + B)C = \begin{bmatrix} (a_{11} + b_{11})c_1 + (a_{12} + b_{12})c_2 + (a_{13} + b_{13})c_3 \ (a_{21} + b_{21})c_1 + (a_{22} + b_{22})c_2 + (a_{23} + b_{23})c_3 \ (a_{31} + b_{31})c_1 + (a_{32} + b_{32})c_2 + (a_{33} + b_{33})c_3 \end{bmatrix} \). This matrix provides the total revenue for each sibling over the entire weekend.

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra used to combine two matrices. It requires specific conditions: the number of columns in the first matrix must equal the number of rows in the second. If matrix \( A \) has dimensions \( m \times n \) and matrix \( B \) is \( n \times p \), their product \( AB \) results in a new matrix of size \( m \times p \). Each element of the resulting matrix is computed as the dot product of corresponding rows of the first matrix and columns of the second.
For example, in our exercise, matrix \( A \) (sales on Saturday) and matrix \( C \) (prices) are multiplied to give a revenue matrix. The entries in the resulting matrix \( AC \) are calculated as:

• Entry 1: \( a_{11}c_1 + a_{12}c_2 + a_{13}c_3 \)
• Entry 2: \( a_{21}c_1 + a_{22}c_2 + a_{23}c_3 \)
• Entry 3: \( a_{31}c_1 + a_{32}c_2 + a_{33}c_3 \)

Each entry represents the total revenue each sibling made on Saturday. Understanding matrix multiplication helps in identifying relationships between data sets, such as sales and revenue in our case.

Matrix Addition

Matrix addition is a straightforward operation used to combine two matrices of the same dimensions by adding their corresponding elements. It's represented as \( C = A + B \), where every element \( c_{ij} \) in the resulting matrix is computed as \( a_{ij} + b_{ij} \).
This operation is intuitive and reflects common scenarios, such as aggregating data over time. In our exercise, matrices \( A \) and \( B \) represent sales data for Saturday and Sunday, respectively. By adding them, we find the total pounds of watermelons, yellow squash, and tomatoes sold by each sibling during the weekend:

• For Amy: \( a_{11} + b_{11} \)
• For Beth: \( a_{21} + b_{21} \)
• For Chad: \( a_{31} + b_{31} \)

This operation gives insights into the overall performance for each sibling over the entire weekend.

Matrices in Algebra

Matrices play a crucial role in algebra and are used for various operations to solve complex linear equations, represent data transformations, and model real-world scenarios. They provide a structured way to handle systems with multiple variables. Matrices can also represent relationships, like our problem of farm produce sales over a weekend.
Breaking down larger problems into matrices allows for easier calculations and modeling in fields like computer science, physics, and economics. For instance, our matrix problem interprets sales data as a manageable collection of numbers, providing insights into total revenues and quantity sold.
By utilizing matrix operations—addition and multiplication—we uncover comprehensive financial results without manually calculating individual entries. This translates real-world data into mathematical language that is easier to analyze and interpret.

Interpretation of Matrix Results

Interpreting matrix results is about making sense of the information that these mathematical objects represent. In our exercise, each matrix operation translates into a meaningful economic insight.
The product \( AC \) or \( BC \) results in a column matrix where each entry represents total revenue generated by each sibling on different days. This helps to identify who sold the most and how much they earned.
Meanwhile, the sum \( A + B \) provides total quantities sold over the weekend, offering insights into inventory management and logistical needs.
Finally, calculating \( (A + B)C \) gives weekend revenue aggregated per sibling, allowing the family to evaluate earnings more effectively. Interpreting these results enables effective decision-making, offering a clearer understanding of the underlying datasets and their real-world implications.