Problem 52 - College Algebra

Question

Describe what is meant by the augmented matrix of a system of linear equations.

Step-by-Step Solution

Answer

An augmented matrix is the matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. In the context of a system of linear equations, it represents the coefficients of the variables and the constants from the right-hand side of the equations.

Step 1: Understanding Linear Equations

A linear equation is a mathematical statement equating the sum of several terms, each a product of a constant and the nth power of a variable, to zero. These equations often represent a line in a graph. A system of linear equations, then, is a collection of one or more linear equations involving the same variables.

Step 2: Introducing Matrix Representation

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In the context of systems of linear equations, the coefficients of the variables can be represented by a matrix. Each row of the matrix corresponds to an equation in the system, and each column corresponds to a variable.

Step 3: Defining the Augmented Matrix

The augmented matrix of a system of linear equations is a matrix in which the rightmost column — separated by a horizontal line in written mathematics, or a vertical line in linear algebra software — represents the constants on the other side of the equals sign in the system. This extends the original coefficient matrix to include these constants, \'augmenting\' it.