Problem 1 - Precalculus: Mathematical for Calculus

Question

State the dimension of the matrix. $$\left[\begin{array}{rr} 2 & 7 \\ 0 & -1 \\ 5 & -3 \end{array}\right]$$

Step-by-Step Solution

Answer

The dimension of the matrix is \(3 \times 2\).

Step 1: Understand the definition of matrix dimensions

Matrix dimensions are defined by the number of rows and columns it has. The general notation is \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.

Step 2: Count the number of rows

In the given matrix \( \begin{bmatrix} 2 & 7 \ 0 & -1 \ 5 & -3 \end{bmatrix} \), observe the number of horizontal lines (rows) that extend across the matrix. There are 3 rows here: \( [2, 7] \), \( [0, -1] \), and \( [5, -3] \).

Step 3: Count the number of columns

For the same matrix, count how many vertical lines (columns) there are in each row. Each row has 2 columns: \( [2, 0, 5] \) and \( [7, -1, -3] \).

Step 4: State the dimension of the matrix

Combine the results from Steps 2 and 3: the matrix has 3 rows and 2 columns. Therefore, the dimension of the matrix is \( 3 \times 2 \).

Key Concepts

Understanding Matrix Rows

Matrix rows are horizontal lines of elements that extend across a matrix.
Each row contains elements that are lined up side by side.
Think of rows as layers that sit horizontally.In our example matrix:\[\begin{bmatrix}2 & 7 \0 & -1 \5 & -3 \\end{bmatrix}\]Here, we have three distinct rows, each represented by a horizontal grouping:

• The first row contains the elements \([ 2, 7 ]\)
• The second row comprises \([ 0, -1 ]\)
• The third row is \([ 5, -3 ]\)

Thinking in terms of rows helps us progress to understanding the dimensions of the matrix.Rows are one part of determining how a matrix is structured, and counting them accurately sets the foundation for defining matrix dimensions.

Exploring Matrix Columns

Matrix columns run vertically through a matrix and organize data into these groups by stacking elements on top of each other.
Just like rows, columns form an essential part of understanding matrix structure.Looking again at our example matrix:\[\begin{bmatrix}2 & 7 \0 & -1 \5 & -3 \\end{bmatrix}\]Here, we examine the vertical stacks:

• The first column includes the elements \([ 2, 0, 5 ]\)
• The second column contains \([ 7, -1, -3 ]\)

Counting the columns gives us the second component needed to determine matrix dimensions. Each element in a column shares its vertical alignment, like a pole of stacked data.Columns complement rows, providing a 2D structure to the data set within a matrix.

Introduction to Matrix Notation

Matrix notation is the way we describe the dimensions of a matrix.
It is typically noted in the form of \( m \times n \), where \( m \) and \( n \) represent rows and columns respectively.In our example:\[\begin{bmatrix}2 & 7 \0 & -1 \5 & -3 \\end{bmatrix}\]By counting the rows and columns, we determine:

• There are 3 rows.
• There are 2 columns.

Thus, the matrix notation for this particular matrix is \( 3 \times 2 \).Matrix notation provides a neat and concise description of the matrix, making communication about the matrix's form easier and more standardized. It is crucial in mathematics and allied disciplines to convey structured data effectively.