Problem 12 - Precalculus with Limits

Question

In Exercises 9-14, determine the order of the matrix. \( \left[\begin{array}{rrrr} -3 & 7 & 15 & 0 \\ 0 & 0 & 3 & 3 \\ 1 & 1 & 6 & 7 \end{array}\right] \)

Step-by-Step Solution

Answer

The order of the matrix is \(3 x 4\).

Step 1: Identify the number of rows

Looking at the given matrix, there are three horizontal lines of numbers, meaning there are three rows.

Step 2: Identify the number of columns

Next, looking at the given matrix, there are four vertical lines of numbers, meaning there are four columns.

Step 3: Formulate the order of the matrix

Finally, since the order of a matrix is given by the number of rows by the number of columns, the order for this matrix is '3 x 4'.

Key Concepts

Matrix Rows

When working with matrices, understanding the concept of "rows" is fundamental. Rows in a matrix are the horizontal arrangements of numbers or elements. Each row is like a separate line of information lying flat within the matrix. Imagine each row as a set of data that might represent different aspects of a dataset. For instance, if you're tracking classroom attendance, each row could stand for daily data, such as the number of students present.

In our example matrix, we observe three rows:

• The first row is \([-3, 7, 15, 0]\)
• The second row is \([0, 0, 3, 3]\)
• The third row is \([1, 1, 6, 7]\)

Rows are always counted from top to bottom, so figuring out the number of rows is straightforward—just count the horizontal lines in the matrix. Understanding the rows is the first step in determining the order of a matrix.

Matrix Columns

Another pivotal element in determining a matrix is the "columns". Columns in a matrix are simply the vertical stacks of numbers. These run from top to bottom, just like columns in a spreadsheet. Each column typically represents a distinct attribute or characteristic of the data.

In the given matrix, there are four columns:

• The first column contains \([-3, 0, 1]\)
• The second column contains \([7, 0, 1]\)
• The third column contains \([15, 3, 6]\)
• The fourth column contains \([0, 3, 7]\)

Counting columns is similar to counting rows—just look at the vertical lines of elements. Keeping track of columns ensures that we can identify each variable or feature they might represent in a dataset. Columns indicate the depth and width of the matrix's dataset structure.

Matrix Dimensions

In the context of matrices, "dimensions" refers to the matrix's size, essentially a product of its rows and columns. The dimension is expressed as rows multiplied by columns, written as \(m \times n\), where \(m\) and \(n\) are integers representing rows and columns, respectively.

In the example provided, the matrix has:

• 3 rows (horizontal)
• 4 columns (vertical)

Hence, its dimensions are \(3 \times 4\), pronounced "three by four." Understanding matrix dimensions is crucial as it determines the nature of operations that can be performed on a matrix, such as addition, multiplication, or finding the inverse. Each matrix's dimensions must be compatible with the matrices it interacts with.

The order or dimension is often considered the "identity" of a matrix, crucial for anyone dealing with linear algebra, as it dictates how the data within the matrix interrelates.