Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Determine the order of the matrix. $$ \left[\begin{array}{rrr} 6 & 4 & 1 \\ 8 & 3 & 0 \\ -1 & 2 & 1 \\ 1 & 5 & 4 \end{array}\right] $$

Step-by-Step Solution

Answer

The order of the given matrix is '4 x 3'

Step 1: Identification of Rows and Columns

In our matrix we have 4 rows: 6 4 1, 8 3 0, -1 2 1, 1 5 4. And we have 3 columns: 6 8 -1 1, 4 3 2 5, 1 0 1 4.

Step 2: Identify the Order

Therefore, the order of the matrix is '4 x 3', where '4' is the number of rows and '3' represents the number of columns in the matrix.

Key Concepts

Matrix Rows

In the world of matrices, understanding rows and columns is key to mastering the basics. A matrix is organized into horizontal lines known as rows. Each row contains a list of numbers, one next to the other.
In our given matrix example:

• The first row is: \( 6 \ 4 \ 1 \)
• The second row is: \( 8 \ 3 \ 0 \)
• The third row is: \( -1 \ 2 \ 1 \)
• The fourth row is: \( 1 \ 5 \ 4 \)

Knowing how to identify and count rows helps you determine part of what is called the 'order' of the matrix. In this case, there are 4 rows total. Each row contributes to the overall structure, giving us a clear foundation for understanding the matrix as a whole.

Matrix Columns

Just like rows, columns are a fundamental part of the matrix structure. Columns run vertically, containing numbers lined up from top to bottom.
In our matrix example:

• The first column is: \( 6 \ 8 \ -1 \ 1 \)
• The second column is: \( 4 \ 3 \ 2 \ 5 \)
• The third column is: \( 1 \ 0 \ 1 \ 4 \)

Counting the number of columns allows you to complete the picture of what the matrix order will be. In this matrix, there are 3 columns, making it easy to visualize and work with the structure presented.

Matrix Dimensions

The concept of matrix dimensions refers to the 'order' of the matrix, expressed as a combination of its rows and columns.
This is written as "rows × columns." Therefore, our matrix, with 4 rows and 3 columns, has dimensions of "4 × 3."
This format conveys two important pieces of information:

• How tall the matrix is (the number of rows)
• How wide the matrix is (the number of columns)

Determining matrix dimensions is essential for any mathematical operations involving matrices, such as matrix addition, multiplication, or even inversion. By understanding dimensions, you gain control over how different matrices relate and interact with each other in various calculations.