Problem 32 - Algebra 2

Question

Anna made a table to show how much money she and two of her friends earned for summer chores. $$\begin{array}{|c|c|c|c|}\hline \text { Anna } & {\$ 20} & {\$ 40} & {\$ 0} \\\ \hline \text { Rob } & {\$ 12} & {\$ 35} & {\$ 40} \\ \hline \text { Carla } & {\$ 15} & {\$ 55} & {\$ 70} \\ \hline\end{array}$$ a. Display the data in Anna's table in a matrix A with each row representing someone's earnings. b. Rob made his own matrix, \(R,\) to show the earnings. \(R=\left[\begin{array}{rrr}{12} & {20} & {15} \\ {35} & {40} & {55} \\ {40} & {0} & {70}\end{array}\right]\). What do values in the first column of matrix \(R\) represent?

Step-by-Step Solution

Answer

a. Anna's earnings matrix A is \begin{bmatrix} 20 & 40 & 0 \ 12 & 35 & 40 \ 15 & 55 & 70 \begin{bmatrix}. b. The first column in Rob's matrix R represents his own earnings for the three different chores.

Step 1: Create Anna's Earnings Matrix

Use the given table to create a matrix, where each row represents the earnings of Anna, Rob, and Carla for summer chores. The first column will represent the earnings for the first chore, the second column for the second chore, and the third column for the third chore.

Step 2: Interpret Rob's Earnings Matrix

Look at matrix R given as Rob's earnings matrix. Each value in the first column represents Rob's claiming of the earnings for some particular chores. We need to interpret which earnings these represent by comparing with the original table.

Key Concepts

Anna's Earnings Matrix

Understanding the construction of Anna's Earnings Matrix is crucial for grasping the fundamentals of matrix representation in algebra. The matrix serves as a streamlined way to organize and display data. In the context of the exercise, Anna has documented the earnings for herself and two friends - Rob and Carla - over three different chores performed during the summer.
Matrix A, which represents Anna's Earnings Matrix, is structured so that each row corresponds to a person, with the three columns signifying the amount earned for each summer chore. Hence, Matrix A would look like this:
\[ A = \begin{bmatrix} 20 & 40 & 0 \ 12 & 35 & 40 \ 15 & 55 & 70 \end{bmatrix} \]
By using this matrix, we can easily glance at an individual's earnings per task and calculate totals or comparisons as needed, which is much more efficient than looking through a standard table.

Rob's Earnings Matrix

Turning our attention to Rob's Earnings Matrix, denoted as matrix R, we notice a different pattern from the structure found in Anna's matrix. Rob's matrix is organized to place emphasis on each chore rather than the individual completing it.
So, matrix R is laid out like this:
\[ R = \begin{bmatrix} 12 & 20 & 15 \ 35 & 40 & 55 \ 40 & 0 & 70 \end{bmatrix} \]

What Does the First Column Represent?

In this configuration, the first column of matrix R houses the amount each person earned from the first chore, with the rows now representing each separate chore. Specifically, Rob earned $12 from the first chore, while the amounts for the second and third chores from the original table are now the second and third rows in matrix R. This example showcases how matrices can be tailored to emphasize different aspects of the data, depending on how they are structured.

Interpreting Matrices

Interpreting matrices is a skill that proves itself indispensable when one engages with algebraic data representation. It enables us to extract meaningful information from a compact grid of numbers, leading to insights that may not be as evident from raw data sets.
To interpret a matrix, we must first understand its structure: which entities represent rows and columns, and what the values inside the matrix signify. In both Anna's and Rob's earnings matrices, the numbers represent money earned from summer chores. However, the matrices differ in their row and column assignments to represent different perspectives.

• In Anna's Earnings Matrix, rows signify individuals, and columns represent distinct chores.
• In Rob's Earnings Matrix, rows indicate chores, while columns convey the earnings of each person for the first chore.

Through these matrices, one can perform various operations such as adding to find totals, comparing to determine who earned more, or even categorizing to highlight patterns. The key takeaway is that matrices are not just a collection of numbers; they tell a story and answer questions about the underlying dataset they represent.