Problem 10 - Algebra 2

Question

State the dimensions of each matrix. $$ \left[\begin{array}{l}{7} \\ {8} \\ {9}\end{array}\right] $$

Step-by-Step Solution

Answer

The matrix is 3x1.

Step 1: Understand the Matrix Structure

The provided matrix is a vertical arrangement of numbers. It consists of a single column with multiple rows.

Step 2: Count the Number of Rows

Count the number of individual elements arranged vertically. Here, there are three numbers: 7, 8, and 9, which means the matrix has 3 rows.

Step 3: Count the Number of Columns

In the given matrix, there is only one vertical alignment of numbers. Therefore, there is just 1 column.

Step 4: State the Dimensions

After identifying the number of rows and columns, state the dimensions of the matrix. The dimensions are given as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns. Thus, the dimensions are \( 3 \times 1 \).

Key Concepts

Matrix Notation

Understanding matrix notation is crucial for working with mathematical problems involving matrices. In matrix notation, a matrix is typically represented by a large capital letter like \( A \) or \( B \). This helps to easily differentiate between various matrices used in equations or systems. Matrices are expressed in a rectangular array where numbers or elements are arranged in rows and columns. The size of a matrix is defined by its dimensions, identified by the number of rows and the number of columns it has. For instance, in a matrix labeled \( A \), if it has 3 rows and 2 columns, it would be written in matrix notation as \( A_{3 \times 2} \). This notation succinctly conveys both the structure and the content of the matrix.When solving problems, recognizing this notation helps you quickly understand the organization of the data within the matrix.

Rows and Columns

The concept of rows and columns is fundamental when dealing with matrices. A row in a matrix is a horizontal series of numbers or elements, whereas a column is a vertical series. In the context of matrices, the rows and columns determine the overall shape and organization of the matrix. Each row is a separate horizontal entity within the matrix, usually starting with the first topmost series. Columns, on the other hand, begin from the leftmost series and extend vertically.

• To determine the rows, count each distinct horizontal line of numbers.
• For columns, count each unique vertical line of numbers.

In our example, the matrix has 3 rows and 1 column. Recognizing this arrangement is key to understanding the characteristics and dimensions of any matrix, which directly influences mathematical operations such as addition or multiplication involving matrices.

Matrix Structure

The structure of a matrix reflects its dimensions, determined by the count of its rows and columns. This structure lays the foundation for various operations and understanding the type of data representation a matrix provides.A matrix can be any size; it could be a small \( 2 \times 2 \) or a very large \( 100 \times 100 \), depending on the context or the problem. It is structured so that every entry or element is defined by its position in terms of row and column. For instance, an element positioned at row 2, column 3 is often denoted as \( a_{23} \).The matrix in question has a simple structure:

• It consists of three separate, distinct rows: \( 7, 8, 9 \).
• It has only one column, meaning all entries are within a single vertical alignment.

Understanding this straightforward arrangement of our example matrix (\( 3 \times 1 \)) is foundational for grasping more complex matrix operations and concepts, ensuring clarity and accuracy in mathematical problem-solving.