Problem 13 - Precalculus with Limits

Question

In Exercises 9-14, determine the order of the matrix. \( \left[\begin{array}{rr} 33 & & 45\\\ -9 & & 20 \end{array}\right] \)

Step-by-Step Solution

Answer

The order of the given matrix is 2x2

Step 1: Identify the rows in the matrix

Look at the matrix and count the number of rows. A row in a matrix is a set of numbers lined horizontally. Each matrix has at least one row. The given matrix has two rows.

Step 2: Identify the columns in the matrix

Similarly, count the number of columns in the matrix. A column in a matrix is a set of numbers lined vertically. Each matrix has at least one column. The given matrix has two columns.

Step 3: Write the order of the matrix

Having identified the number of rows and columns, we can say that the order of the matrix is 'Number of Rows' x 'Number of Columns'. For our given matrix, the order is 2x2.

Key Concepts

Matrix Rows

Matrix rows are sets of numbers arranged horizontally in a matrix. Think of them like rows of seats in a theater where each row sits side by side. In matrices, rows are crucial in determining the structure of the data. Here are some key points to help you visualize and understand matrix rows:

• The position of a row is always horizontal.
• Every matrix will have at least one row, even if it's a single set of numbers.
• To count rows, look horizontally and see how many sets of numbers are stacked on top of each other.

In our original exercise, the matrix is:\[\begin{bmatrix} 33 & 45\-9 & 20\end{bmatrix}\]In this example, there are two distinct horizontal lines of numbers, which means there are two rows in the matrix.

Matrix Columns

Matrix columns are a vertical set of numbers in a matrix. Visualize them like columns in a building, holding up each floor in the data layout. Columns help to organize and separate different sets or categories of data. Here are some key insights about matrix columns:

• Columns always appear vertically within the matrix.
• Just like rows, every matrix should have at least one column.
• Columns are counted by looking vertically from top to bottom, noting how many sets are next to each other.

In the same matrix example as before:\[\begin{bmatrix} 33 & 45\-9 & 20\end{bmatrix}\]We see two vertical arrangements of numbers, indicating that there are two columns in the matrix.

Matrix Notation

Matrix notation is a standardized way of representing the dimensions and content of a matrix. This helps quickly and clearly present information about the matrix's structure and size. Here's how matrix notation works:

• A matrix is generally denoted by a capital letter, like \(A\).
• The order of the matrix is noted as 'Rows x Columns'. This means listing the number of rows first and then the number of columns. In notation, it clarifies size and layout.
• A matrix with 2 rows and 2 columns would be written as a 2x2 matrix.

Returning to our matrix example, the structure:\[\begin{bmatrix} 33 & 45\-9 & 20\end{bmatrix}\]would be notated as a 2x2 matrix because it consists of 2 rows and 2 columns. This notation is important for identifying compatibility with matrix operations such as addition and multiplication.