Augmented Matrices - Varsity Tutors

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Augmented Matrices

Study Guide

Augmented Matrices

An augmented matrix is a matrix derived from a system of linear equations. It is a convenient and compact way to represent and work with systems of linear equations.

Important Notes

• •Augmented matrices can be used to solve systems of linear equations.
• •The augmented matrix of a system of linear equations is obtained by writing the coefficients of the variables and constants in columns.
• •The augmented matrix does not include the variables in the system of equations.
• •Each row in an augmented matrix represents an equation in the system.
• •Each of the first n columns represents the coefficients of the variables, and the last column represents the constants of the equations.

Mathematical Notation

$Ax = b$ represents a system of linear equations.$[A | b]$ represents the augmented matrix of a system of linear equations.$A$ represents the matrix of coefficients.$x$ represents the column vector of variables.$b$ represents the column vector of constants.In the context of augmented matrices, the vertical bar in the matrix separates the coefficients of the variables from the constants.

Why It Works

Augmented matrices work because they represent the same information as a system of linear equations, but in a more compact form. This makes it easier to perform operations on the equations and solve the system.

Remember

Remember that the order of the rows and columns in an augmented matrix corresponds to the order of the equations and variables in the system of equations.

Quick Reference

Augmented Matrix:$[A | b]$

Understanding Augmented Matrices

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Beginner Explanation

An augmented matrix is derived from a system of linear equations by writing the coefficients of the variables and constants in columns.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

What is the augmented matrix of the system of equations $x + y = 5$ and $2x - y = 3$?

A$\begin{bmatrix}1 & 1 & 5\\2 & -1 & 3\end{bmatrix}$B$\begin{bmatrix}1 & 2 & 5\\1 & -1 & 3\end{bmatrix}$C$\begin{bmatrix}1 & 1 & 2\\5 & 3 & -1\end{bmatrix}$D$\begin{bmatrix}5 & 1 & 1\\3 & 2 & -1\end{bmatrix}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Pizza Shop Scenario

A pizza shop sells two types of pizzas: cheese and pepperoni. The cheese pizza costs $10 and the pepperoni pizza costs $12. One day, they sold a total of 50 pizzas and made $580. How many of each type of pizza did they sell?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

What does each row in an augmented matrix represent?

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Challenge Quiz

Single Choice QuizAdvanced

What is the solution to the system of equations represented by the augmented matrix $[1 2 | 3; 2 5 | 8; 3 7 | 11]$?

A$x = -1, y = 2$B$x = 1, y = 1$C$x = 0, y = 0$DThe system has no solutionCheck AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

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