Problem 15 - Algebra 2

Question

Use the table below $$\begin{array}{|c|c|c|c|c|c|}\hline 1998 & {1999} & {2000} & {2001} & {2002} & {2003} \\ \hline 847.0 & {938.9} & {942.5} & {881.9} & {803.3} & {745.9} \\\ \hline 0.5 & {2.5} & {3.3} & {7.9} & {10.7} & {17.5} \\\ \hline\end{array}$$ Display the data in a matrix \(A\) with columns representing years. Identify \(a_{23}\) and tell what it represents.

Step-by-Step Solution

Answer

Element a_{23} of matrix A is 3.3, representing the percentage growth in revenue for the year 2000.

Step 1: Understand the data structure

First, understand how the table is structured. The table has two rows of data, each corresponding to different types of information. The first row represents the revenue (in billions) for each year from 1998 to 2003. The second row gives the percentage growth over the previous year's revenue for the same years.

Step 2: Construct the matrix A

Next, create a matrix 'A' with each column representing a year from 1998 to 2003. The matrix will have two rows: the first row for the revenue data and the second row for the percentage growth data.

Step 3: Identify the element a_{23}

The element 'a_{23}' is located in the second row and the third column of matrix A. This corresponds to the year 2000, and since it's in the second row, it represents the percentage growth for that year.

Key Concepts

Algebraic Matrices

Algebraic matrices are a fundamental concept in mathematics, often used to represent and manipulate sets of numbers and variables in a rectangular array format. In essence, a matrix is a collection of numbers arranged into fixed rows and columns, making them essential tools for various applications ranging from computer graphics to quantum mechanics.

When it comes to exercises like the one provided, matrices can be particularly useful for organizing and displaying data in a structured way. In the context of this problem, we're using a matrix to represent financial data over a series of years. The matrix, denoted as \( A \), encapsulates two types of data across its two rows: the first for revenue figures and the second for percentage growth rates.

For improved understanding, it's crucial to grasp that each element of the matrix, indicated by \( a_{ij} \), where \( i \) represents the row number and \( j \) represents the column number, corresponds to a specific piece of data. When referring to matrix elements, always ensure to start counting from one, as opposed to zero, which is common in many programming languages.

Data Interpretation

Interpreting data correctly is invaluable for drawing meaningful conclusions from numbers and statistics. In our matrix example, data interpretation involves analyzing the numbers in the matrix to uncover trends, identify patterns, and understand the health and performance of an entity over time.

For instance, by examining the matrix \( A \) from our exercise, one can interpret the financial growth of a company across the years. The second row of the matrix, which gives the percentage growth, is particularly interesting for insights into year-over-year performance. By looking at an element like \( a_{23} \), which we identify as 3.3, we learn that in the year 2000, the company experienced a 3.3% growth compared to the previous year.

When interpreting matrices, remember that the context provided by the matrix's structure is just as important as the numbers themselves. Each figure conveys different information depending on its position within the matrix. Encouraging students to focus on the meaning behind numbers and their relative positions will enhance their data interpretation skills and enable them to conduct more nuanced analysis.

Percentage Growth Calculation

Percentage growth calculation is a vital concept in businesses and economics as it measures change over time, reflecting increases or decreases. It's essentially a ratio that compares the difference between the old and new values to the old value's magnitude, expressed as a percent.

In our exercise, percentage growth is presented for successive years, indicating how well a company is scaling its revenue. Calculating percentage growth can be performed using the formula:
\[ \text{Percentage Growth} = \left(\frac{\text{New Value} - \text{Old Value}}{\text{Old Value}}\right) \times 100 \% \]
With data presented in a matrix, as we have in the problem, these calculations can be easily visualized and computed, if needed. Students might be required to not only read these values but also understand how they were derived or even calculate them given raw data.

Emphasizing the importance of identifying the base year for comparison, the value being compared, and the appropriate placement of these values in the calculation will support students in accurately determining percentage growth rates. Moreover, encouraging students to practice these calculations helps to cement their understanding of growth trends in a variety of contexts.