Problem 2 - A Graphical Approach to Precalculus with Limits

Question

Find the dimension of each matrix. Identify any square, column, or rove matrices. Do not use a calculator. $$\left[\begin{array}{rrr}2 & -8 & 6 \\ 1 & 0 & -5 \\ 5 & -2 & 3\end{array}\right]$$

Step-by-Step Solution

Answer

The matrix has dimension \(3 \times 3\) and is a square matrix.

Step 1: Identify the rows of the matrix

Examine the matrix and count the number of horizontal arrays of numbers. The given matrix has three horizontal rows: \([2, -8, 6]\), \([1, 0, -5]\), and \([5, -2, 3]\). Thus, the matrix has 3 rows.

Step 2: Identify the columns of the matrix

Count the number of vertical arrays of numbers. The matrix has three vertical columns: \(\begin{bmatrix} 2 \ 1 \ 5 \end{bmatrix}\), \(\begin{bmatrix} -8 \ 0 \ -2 \end{bmatrix}\), and \(\begin{bmatrix} 6 \ -5 \ 3 \end{bmatrix}\). Thus, the matrix has 3 columns.

Step 3: Determine the dimension

The dimension of a matrix is given as the number of rows by the number of columns. Therefore, the dimension of this matrix is \(3 \times 3\).

Step 4: Identify the type of matrix

Since the number of rows is equal to the number of columns (3 rows and 3 columns), this matrix is classified as a square matrix. Furthermore, the definition of row and column matrices requires the dimension to be either \(1 \times n\) or \(n \times 1\), which does not apply here.

Key Concepts

Square Matrix

A square matrix is a special type of matrix where the number of rows equals the number of columns. In the matrix provided in the original exercise, there are 3 rows and 3 columns, giving it a shape of 3 by 3. This characteristic makes it a square matrix.
A square matrix is often denoted by the symbol \( n \times n \), where \( n \) is the number of rows (or columns since they are equal) and is referred to as the dimension of the matrix.
A square matrix has unique properties that distinguish it from other types of matrices, such as:

• Diagonal elements: These are located from the top-left to the bottom-right of the matrix and often hold special significance, such as in trace calculations, where the sum of diagonal elements is considered.
• Symmetry: A square matrix can be symmetric, meaning that it is equal to its transpose. That is, the element in the \( i^{th} \) row and \( j^{th} \) column is equal to the element in the \( j^{th} \) row and \( i^{th} \) column.
• Determinant: Only square matrices have determinants, which can be a crucial function in calculus and linear algebra, used to determine if a matrix is invertible.

Rows and Columns

Understanding rows and columns in a matrix is foundational to grasping matrix dimensions. In a matrix, rows are horizontal entities, while columns are vertical ones.
For example, in the matrix given:

• There are three rows:
  • The first row contains [2, -8, 6]
  • The second row contains [1, 0, -5]
  • The third row contains [5, -2, 3]
• Likewise, there are three columns:
  • The first column contains \( \begin{bmatrix} 2 \ 1 \ 5 \end{bmatrix} \)
  • The second column contains \( \begin{bmatrix} -8 \ 0 \ -2 \end{bmatrix} \)
  • The third column contains \( \begin{bmatrix} 6 \ -5 \ 3 \end{bmatrix} \)

This structure of rows and columns helps determine the dimension of a matrix, expressed as "rows by columns" (e.g., \( 3 \times 3 \) here). Remembering that rows traverse horizontally and columns vertically can help you easily visualize and manipulate matrices in your mind.

Matrix Types

Matrices can come in various forms, each serving different purposes and applications in mathematics. Understanding these types will help in recognizing their properties and uses:

• Row Matrix: This is a type of matrix with only one row. Dimensions of a row matrix are \( 1 \times n \), where \( n \) is the number of columns.
• Column Matrix: Opposite of a row matrix, a column matrix has only one column. It has dimensions of \( n \times 1 \), where \( n \) is the number of rows.
• Square Matrix: As already discussed, a square matrix has the same number of rows and columns. It plays a pivotal role in linear algebra, especially in solving equations and transformations.

These basic matrix types pave the way to more complex concepts like diagonal matrices, zero matrices, and identity matrices. Each has unique properties and particular areas of application, making them valuable tools in mathematics and beyond. Recognizing the type of matrix you are working with is the first step towards understanding its potential uses and operations.