Problem 5 - Algebra and Trigonometry

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: Identify the Matrix

The given matrix is a rectangular array of numbers: \(\left[\begin{array}{rr}{2} & {7} \ {0} & {-1} \ {5} & {-3}\end{array}\right] \) This matrix has a specific layout that we need to understand to determine its dimensions.

Step 2: Determine the Number of Rows

Count how many horizontal lines of entries (rows) are in the matrix. In the given matrix, the rows are: 1. \([2, 7]\) 2. \([0, -1]\) 3. \([5, -3]\) Therefore, there are 3 rows.

Step 3: Determine the Number of Columns

Count how many vertical lines of entries (columns) are in the matrix. In the given matrix, the columns are: 1. \(\begin{array}{c}{2} \ {0} \ {5}\end{array}\) 2. \(\begin{array}{c}{7} \ {-1} \ {-3}\end{array}\) Thus, there are 2 columns.

Step 4: State the Matrix Dimension

The dimension of a matrix is given by the number of rows by the number of columns, typically written as 'number of rows' \(\times\) 'number of columns'. For this matrix, the dimension is \(3 \times 2\).

Key Concepts

Rows and Columns

In the world of matrices, understanding the concept of rows and columns is fundamental. Think of a matrix as a box-like structure filled with numbers. Within this box, we have horizontal strips called rows and vertical strips known as columns. Rows are the horizontal sequences of numbers in a matrix. For example, if we take the matrix \[\left[\begin{array}{rr}2 & 7 \0 & -1 \5 & -3\end{array}\right]\]We can see that there are three rows. Each row is a set of numbers aligned horizontally:

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

These rows give us the first part of a matrix's dimension. Columns, on the other hand, are vertical sequences within a matrix. For the same matrix, the columns line up as follows:

• The first column is \(\begin{array}{c}2 \ 0 \ 5\end{array}\)
• The second column is \(\begin{array}{c}7 \ -1 \ -3\end{array}\)

Together, the number of rows and columns help determine the matrix's "dimension." Thus, this matrix has 3 rows and 2 columns.

Matrix Structure

A matrix is not just a random collection of numbers. It's an organized, rectangular array that follows a specific format. This structure is key to performing operations like addition, multiplication, and finding determinants or inverses. In mathematical terms, the matrix we are examining is:\[\left[\begin{array}{rr}2 & 7 \0 & -1 \5 & -3\end{array}\right]\]This matrix is a 3x2 matrix. The rows and columns create the fundamental framework of the matrix, helping it maintain its form. To understand the structure:

• All entries in a matrix are organized systematically in rows and columns.
• Each entry, or element, is located by "coordinates" referencing its row and column (like in a grid).
• The intersection of a specific row and column locates an individual element, such as `2` which is in the first row and first column.

This predefined structure makes matrices both powerful and logical tools in mathematics.

Array of Numbers

At its core, a matrix is an array of numbers that conveys information in a compact, visual format. This array is arranged into rows and columns, just like a chessboard has squares, each filled with valuable data. Taking our example matrix:\[\left[\begin{array}{rr}2 & 7 \0 & -1 \5 & -3\end{array}\right]\]The numbers \(2, 7, 0, -1, 5,\) and \(-3\) are all part of this array. In mathematics, organizing data in arrays allows for operations such as:

• Addition: Adding matrices involves adding corresponding entries.
• Multiplication: Multiplying matrices involves combining the arrays in prescribed ways.
• Other calculations like transposing or finding determinants (for square matrices).

By understanding a matrix as an array of numbers, we unlock its potential to model and solve complex problems in fields ranging from physics to economics. This makes the array structure a fundamental aspect of why matrices are so widely used and trusted in various applications.