Problem 9 - Precalculus: Mathematics for Calculus

Question

State the dimension of the matrix. $$\left[\begin{array}{lll} 1 & 4 & 7 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is 1x3.

Step 1: Identify the rows in the matrix

Look at the matrix provided:\[\begin{array}{lll}1 & 4 & 7\end{array}\]This matrix is presented in a single line. Therefore, it has 1 row.

Step 2: Identify the columns in the matrix

Each number in the row is separated by spaces or commas which form distinct columns.In this matrix, the numbers are distributed over three positions:\[1, 4, 7\]Thus, it has 3 columns.

Step 3: Determine the dimension of the matrix

The dimension of a matrix is determined by the number of rows and columns it has.Since this matrix has 1 row and 3 columns, its dimension is written as \(1 \times 3\).

Key Concepts

Matrix Rows

When looking at a matrix, the first step in identifying its dimensions is to check the number of matrix rows it contains. A row in a matrix is defined as a horizontal arrangement of elements. Each row contains values that can range from numbers to variables. In our given example:\[\begin{array}{lll}1 & 4 & 7\end{array}\]You will notice all elements are aligned horizontally in one straight line, indicating that there is just one row.
The number of rows is always counted from top to bottom, much like reading a book.
In this matrix, the solitary line of numbers means the matrix has a total of **1 row**.

Matrix Columns

Next, we need to determine the number of matrix columns, which run vertically in the matrix. Each column in a matrix is essentially an alignment of elements arranged from top to bottom in the same vertical line.

Consider our example matrix:\[\begin{array}{lll}1 & 4 & 7\end{array}\]Here, the different numbers are presented side by side, resulting in separate vertical partitions. Think of columns like tree trunks lined up in a row. - The number 1 is the first column.- The number 4 lies in the second column.- The number 7 falls in the third column.
Adding them up, we recognize a total of **3 columns** in this matrix.

Matrix Identification

Once you have counted the number of rows and columns, matrix identification is all about noting down its dimension using a simple format. The dimensions of a matrix specify how many rows and columns it contains, written as \(\text{rows} \times \text{columns}\). For instance, our matrix:\[\begin{array}{lll}1 & 4 & 7\end{array}\]With 1 row and 3 columns, the dimension can be specified as \(1 \times 3\). - This dimension tells us that the matrix spreads across one horizontal line and vertically stretches over three positions.
This method of matrix identification is crucial when working with larger systems, allowing you to quickly ascertain its structure and potential applications.