Problem 20 - Algebra 2

Question

State the dimensions of each matrix. Identify the indicated element. \(\left[\begin{array}{rrr}{4} & {6} & {5} \\ {2} & {-3} & {-7} \\ {1} & {0} & {9}\end{array}\right], a_{23}\)

Step-by-Step Solution

Answer

The matrix is a 3x3 matrix. Element a_{23} is -7.

Step 1: Determine the dimensions of the matrix

Count the number of rows and columns in the matrix. The dimensions of a matrix are given as 'rows' by 'columns'.

Step 2: Identify the element at position a_{23}

Locate the element in the 2nd row and 3rd column of the matrix. The subscript '23' refers to the position of the element within the matrix.

Key Concepts

Elements of a Matrix

An understanding of the elements of a matrix is crucial in various fields of mathematics and applications like computer science and engineering. Fundamentally, a matrix is a collection of numbers arranged in rows and columns that can be used to represent a set of equations or transformations.

The individual numbers in a matrix are referred to as the elements or entries of the matrix. Typically, we denote these elements with a lowercase letter and a subscript that corresponds to the element's position within the matrix. For example, in the matrix exercise provided, the element denoted as a_{23} refers to the element located in the 2nd row and the 3rd column.

To aid in understanding, consider the matrix as a grid where each cell holds a value. When you are asked to identify an element such as a_{23}, you are essentially being asked to find the value that is in the grid's 2nd horizontal row and 3rd vertical column.

Rows and Columns in Matrices

The structure of a matrix is defined by its rows and columns. Rows run horizontally and are usually represented by the first subscript of a matrix's element. Columns run vertically and are identified by the second subscript of the element. Viewing a matrix as a table with horizontal and vertical lines can make the concept easier to grasp.

In any matrix, the order in which the rows and columns are counted is important. The counting starts from the top-left corner and is always horizontal first (rows), then vertical (columns). For instance, if you're given a matrix and asked to state its dimensions, you start by counting the number of horizontal rows and then the number of vertical columns.

Sometimes, you will hear mathematicians refer to these dimensions in terms of ‘m by n’ where ‘m’ stands for the number of rows and ‘n’ for the number of columns. This nomenclature is pivotal in operations such as addition, subtraction, and multiplication of matrices, where the compatibility of dimensions is a pre-requisite for these operations to be performed.

Matrix Notation

Mastering the notation of matrices is essential for communicating and solving matrix-related problems effectively. Matrix notation provides a compact way to present and work with multiple numerical values at once.

A standard notation for a matrix is an uppercase letter, such as A, and its elements are expressed with a corresponding lowercase letter with subscripts, like a_{ij} where i and j indicate the row and column, respectively. In educational texts and problem sets, matrices are often written within square brackets or parentheses with rows delimited by semicolons or line breaks.

The dimension of a matrix is given right next to its name, for instance, A is an m x n matrix, where m is the number of rows and n is the number of columns. The precise order of rows and columns is imperative for understanding the capabilities and limitations of matrix operations, as the structure directly affects the outcome of these operations.