Problem 3 - College Algebra

Question

In Exercises \(1-4\) a. Give the order of each matrix. b. If \(A=\left[a_{i j}\right],\) identify \(a_{32}\) and \(a_{23}\) or explain why identification is not possible. $$ \left[\begin{array}{rrrr} 1 & -5 & \pi & e \\ 0 & 7 & -6 & -\pi \\ -2 & 1 & 11 & -1 \end{array}\right] $$

Step-by-Step Solution

Answer

The order of the matrix is 3x4. Element \(a_{32}\) is 1, and element \(a_{23}\) is -6.

Step 1: Determining the Order of the Matrix

In order to determine the order of the matrix, count the number of rows and the number of columns. The matrix given is a 3x4 matrix, as it has 3 rows and 4 columns

Step 2: Identifying the Elements

Next is to identify the elements. Element \(a_{32}\) is the element at the third row and second column, which is \(1\). Element \(a_{23}\) is the element at the second row and third column, which is \(-6\)

Key Concepts

Matrix Elements Identification

Understanding how to identify elements within a matrix is key to working with matrices in mathematics. Each element in a matrix is specified by two indices: the row number and the column number. This system of indexing starts with the element in the top-left corner as element 11, moving across the row for subsequent columns and down for subsequent rows.

For instance, in the given 3x4 matrix, to find the element labeled as \(a_{32}\), you look at the intersection of the third row and the second column. Similarly, to locate \(a_{23}\), you would find where the second row and the third column cross paths. In our example, \(a_{32}\) corresponds to the number 1, and \(a_{23}\) corresponds to -6. If a requested element has indices outside of the matrix dimensions, identification is not possible because the element does not exist.

Matrix Dimensions

The dimensions of a matrix, commonly referred to as its 'order', are defined by the number of rows and columns it contains. This is usually represented as 'rows x columns'. A 3x4 matrix, such as the one in our exercise, means there are 3 rows and 4 columns.

To correctly determine the order of any matrix, simply count the rows and then the columns. The first number of the matrix order represents the rows, and the second represents the columns. Understanding matrix dimensions is crucial for performing matrix operations like addition, subtraction, multiplication, and finding determinants, as certain operations are only possible with matrices of specific dimensions. It's also important to remember that the elements of matrices are typically denoted in relation to their position within these dimensions.

Matrix Notation

Matrices are represented by capital letters, and their elements are typically denoted by a lowercase letter with two subscript indices, such as \( a_{ij} \), where 'i' represents the row and 'j' represents the column. The notation for a matrix element is a concise way to refer to the position of that element within the matrix.

For example, when looking at the matrix in our exercise, the notation \(A=\big[a_{ij}\big]\) indicates that the matrix is named 'A' and the elements inside it are to be identified by their row-column position. Remember that when referencing elements of a matrix with this notation, the first subscript refers to the row number and the second to the column number, starting with the indices at 1. This notation is foundational for articulating and solving many linear algebra problems.