Problem 23 - Algebra 1

Question

Write and label a matrix to organize the information. Music Store Inventory: cos: 52 sale price titles, 3300 regular price titles Tapes: 28 sale price titles, 1600 regular price titles

Step-by-Step Solution

Answer

The completed and labeled matrix representing the music store inventory is: \[\begin{array}{c|cc} & \text{Sale Price Titles} & \text{Regular Price Titles} \ \hline \text{CDs} & 52 & 3300 \ \text{Tapes} & 28 & 1600 \end{array}\].

Step 1: Define the Rows and Columns of the Matrix

In this problem, the rows of the matrix will be defined by the type of music: CDs and Tapes. The columns will be defined by the price status of the music: sale price titles and regular price titles.

Step 2: Enter the Data

The data given in the problem corresponds to the number of titles for each type of music and price status. For CDs there are 52 sale price titles and 3300 regular price titles. For Tapes, there are 28 sale price titles and 1600 regular price titles.

Step 3: Create the Matrix

In writing the matrix, the first row will represent CDs and the second row will represent Tapes. The first column will represent sale price titles and the second column will represent regular price titles. Thus, the matrix becomes: \[\left[\begin{array}{cc}52 & 3300 \28 & 1600\end{array}\right]\]. This matrix is read as follows: CDs have 52 sale price titles and 3300 regular price titles. Tapes have 28 sale price titles and 1600 regular price titles.

Step 4: Label the Matrix

The final step is to label the matrix according to the type of music and price status. The completed and labeled matrix becomes: \[\begin{array}{c|cc} & \text{Sale Price Titles} & \text{Regular Price Titles} \ \hline \text{CDs} & 52 & 3300 \ \text{Tapes} & 28 & 1600 \end{array}\].

Key Concepts

Matrix Representation

Matrices are a powerful tool used in mathematics to organize and represent data in a structured manner.
Think of a matrix as a grid that helps you see all your information at a glance. Each spot within the grid, called an element, holds a piece of data.

In this example, we represented the inventory of a music store using a matrix. Here, we want to organize how many titles are available at a sale price and how many at a regular price, both for CDs and tapes.
The matrix is set up in two parts:

• Rows: Represent the type of music (CDs and Tapes)
• Columns: Denote the price status (sale price and regular price)

This setup helps us easily view the different combinations of music type and price. Each element in the matrix corresponds to a specific count of titles for a music type and price status.

By using such a matrix, we can quickly and efficiently organize complex sets of data. It makes a large collection of information easier to read, analyze, and compare.

Matrix Labeling

Labeling a matrix is crucial for understanding the data it contains and how it is organized. Proper labels guide the viewer through the matrix and clarify the context of the data.

In our music store inventory matrix, labels help distinguish between different musical types and price statuses. The labels are added outside the matrix to show what each row and column represents:

• Rows are labeled with the type of musical item, "CDs" and "Tapes." This tells us what kind of items are represented in each layer of the matrix.
• Columns are labeled by "Sale Price Titles" and "Regular Price Titles," defining what pricing category the data falls into.

Having these labels allows users to immediately see what kind of data is in each position without needing to refer back to any explanations or legends. This clarity is especially important when working with larger matrices or when more detailed analysis is needed.
Labeling ensures that data remains understandable even as it gets more intricate, making it possible to draw insights swiftly.

Data Organization

Data organization refers to arranging data in a systematic way for easy access and analysis. Matrices are a fundamental tool in organizing data efficiently, especially when you have multiple variables to track.

In the case of the music store inventory, using a matrix simplifies how the information is structured. You can instantly access and interpret key facts like:

• Total sale price titles for CDs and tapes
• Comparison of regular price titles across different music types

Such organization becomes invaluable when handling larger datasets or more complex inventory systems. The structured layout allows data analysts to perform operations like addition, subtraction, and comparisons seamlessly within the matrix framework.

With well-organized data, the process of updating or modifying information becomes less cumbersome. You can locate and amend data points without sifting through clutter. Thus, matrices not only streamline analysis but also enhance decision-making processes.