Problem 6 - Precalculus Mathematics for Calculus

Question

State the dimension of the matrix. $$\left[\begin{array}{rrrr} -1 & 5 & 4 & 0 \\ 0 & 2 & 11 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix has dimension 2x4.

Step 1: Understand the Matrix Layout

The matrix is presented in a standard rectangular arrangement, with numbers organized in rows and columns. To understand its dimensions, we need to count how many rows and columns it has.

Step 2: Count the Rows

Look at the matrix and count the horizontal lines of numbers (rows). This matrix has two horizontal lines of numbers: -1, 5, 4, 0 and 0, 2, 11, 3. Thus, it has 2 rows.

Step 3: Count the Columns

Look at the matrix and count the vertical lines of numbers (columns). Each row has the same number of entries, which are arranged vertically. There are four vertical alignments of numbers; hence, there are 4 columns.

Step 4: State the Dimensions

The dimension of a matrix is given by the number of rows followed by the number of columns. For this matrix, we've identified 2 rows and 4 columns, so its dimension is 2x4.

Key Concepts

Rows and Columns

Understanding the structure of a matrix begins with recognizing its rows and columns. Rows are horizontal lines of numbers, while columns are vertical lines of numbers. This fundamental organization helps us describe and work with matrices in a structured way. For example, consider a matrix with the layout:\[ \begin{array}{cccc} -1 & 5 & 4 & 0 \ 0 & 2 & 11 & 3 \end{array} \]In this case, we have:

• Rows: Each row is a single, horizontal sequence of numbers. Here, the first row is \([-1, 5, 4, 0]\), and the second is \([0, 2, 11, 3]\).
• Columns: A column is a single, vertical sequence. In our matrix, the first column is \([-1, 0]\), the second column is \([5, 2]\), the third column is \([4, 11]\), and the fourth column is \([0, 3]\).

Recognizing these groups is key in understanding and manipulating matrices.

Matrix Representation

Matrix representation refers to how we visually and symbolically express matrices using brackets and numbers. Matrices are an organized system of numbers displayed in rows and columns, enclosed within brackets. This symbolic representation allows us to easily convey its structure and values. For example, a matrix can be represented as:\[ \left[ \begin{array}{cccc} -1 & 5 & 4 & 0 \ 0 & 2 & 11 & 3 \end{array} \right] \]Here’s what to note about its representation:

• The outer brackets show where the matrix begins and ends.
• The numbers are placed in a fixed array of rows and columns.
• Each number or "element" of the matrix is located at a specific position based on its row and column index.

Matrix representation is crucial because it allows mathematicians and scientists to solve complex problems through simplified graphical illustrations.

Matrix Size

The size of a matrix, also referred to as its dimensions, is denoted by the number of rows and columns it contains. This measurement is crucial as it conveys how data is organized within the matrix and can influence how calculations and transformations are performed. For any given matrix, such as:\[ \left[ \begin{array}{cccc} -1 & 5 & 4 & 0 \ 0 & 2 & 11 & 3 \end{array} \right] \]The size is found by counting:

• The number of rows, which in this example, is 2.
• The number of columns, which is 4 in this case.

Thus, the size of the matrix is expressed as \(2 \times 4\). Determining the matrix size is essential for tasks like adding, subtracting, or multiplying matrices, where conforming sizes are necessary to perform operations.