/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 52 Describe what is meant by the au... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 52

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

Defining the Augmented Matrix

The augmented matrix of a system of linear equations is a matrix in which the rightmost column 鈥� separated by a horizontal line in written mathematics, or a vertical line in linear algebra software 鈥� 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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises \(17-26,\) let $$ A=\left[\begin{array}{rr} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{array}\right] \text { and } B=\left[\begin{array}{rr} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{array}\right] $$ Solve each matrix equation for \(X\). $$ 4 A+3 B=-2 X $$

Use Cramer's rule to solve each system. $$ \begin{aligned}x+y+z &=0 \\\2 x-y+z &=-1 \\\\-x+3 y-z &=-8\end{aligned} $$

Determinants are used to show that three points lie on the same line (are collinear). If $$ \left|\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\\x_{3} & y_{3} & 1\end{array}\right|=0 $$ then the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) are collinear. If the determinant does not equal 0, then the points are not collinear. Use this information to work. Are the points \((-4,-6),(1,0),\) and \((11,12)\) collinear?

Evaluate each determinant. $$ \left|\begin{array}{rrrr}1 & -3 & 2 & 0 \\\\-3 & -1 & 0 & -2 \\\2 & 1 & 3 & 1 \\\2 & 0 & -2 & 0\end{array}\right| $$

Use Cramer's rule to solve each system. $$ \begin{aligned}4 x-5 y-6 z &=-1 \\\x-2 y-5 z &=-12 \\\2 x-y &=7\end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.