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Chapter 8: Problem 52

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

Short Answer

Expert verified
The augmented matrix of a system of linear equations is a way of representing the system in a compact matrix form (a rectangular array of numbers). Each row of the matrix corresponds to a linear equation and each column corresponds to a coefficient of a variable. This matrix includes both the coefficients of the variables and the constants or the vectors on the other side of the equation.

Step by step solution

01

Understanding Linear Systems

The first step is to understand what a system of linear equations is. It's basically two or more linear equations that are related to each other in a certain way, usually involving the same variables.
02

Matrix Representation

The next step is understanding the matrix representation of these linear systems. A matrix is a rectangular array of numbers. The linear system can be represented in a compact form using a matrix where each row of the matrix corresponds to a linear equation and each column corresponds to a coefficient of a variable.
03

Defining Augmented Matrix

The augmented Matrix is a form of matrix that includes both the coefficients of the variables and the constants or the vectors on the other side of the equation (after the equals sign). It merges the coefficient matrix and the column of constants into one matrix separated by a line. This makes it easier to use certain operations to solve the linear system.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are mathematical statements that show the relationship between variables and constants. These relationships are expressed in the simplest form with no exponents higher than one.
The general form of a linear equation in one variable is \( ax + b = 0 \), where \( a \) and \( b \) are constants. In more than one variable, such as \( ax + by = c \), the equation represents a straight line in a two-dimensional space.
  • The coefficients \( a \) and \( b \) determine the slope and intersection of the line with the axes.
  • The solution of the linear equation is a set of values for the variables that make the equation true.
A system of linear equations consists of two or more of these linear equations that share common variables.
Matrix Representation
Matrices offer a compact and organized way to handle systems of equations. A matrix is essentially a grid of numbers arranged into rows and columns.
When representing linear equations in a matrix, each row corresponds to an equation, and each column corresponds to the coefficient of a variable in those equations.
  • If you have a system of equations, each equation becomes a row.
  • The number of variables in the system determines the number of columns.
This method of matrix representation simplifies calculations and is crucial for efficient computation in solving systems of linear equations.
System of Linear Equations
A system of linear equations consists of two or more equations that are considered concurrently. These equations often have common unknowns.
In solving these systems, the goal is to find out the values of the variables that satisfy all equations simultaneously.
  • An example is having two equations like \( 2x + 3y = 5 \) and \( 3x + 4y = 7 \).
  • The graphical representation involves plotting straight lines; the solution is where the lines intersect each other.
These solutions can be found using methods like substitution, elimination, or matrix methods such as Gaussian elimination.
Coefficient Matrix
The coefficient matrix is derived directly from the system of linear equations. This matrix is composed solely of the coefficients of the variables in the equations.
For example, in the system \( 2x + 3y = 5 \) and \( 4x - y = 6 \), the coefficient matrix would be:
\[\begin{bmatrix} 2 & 3 \4 & -1 \end{bmatrix}\]
This matrix does not include the constants from the right side of the equations; it's focused solely on the factors of the variables involved.
Identifying and using the coefficient matrix makes it easier to analyze and apply matrix operations to solve the system of equations.
Matrix Operations
Matrix operations involve various techniques that are used to manipulate and solve matrices, including those representing systems of linear equations.
Common operations include matrix addition, subtraction, multiplication, and finding inverses of matrices.
  • Gaussian elimination is a popular method used for solving systems of equations. It involves performing row operations to simplify the system to one that is easily solvable.
  • The operations focus on transforming the matrix to a more convenient form, such as a diagonal or triangular matrix, to easily identify solutions.
Understanding these operations is fundamental when using matrices to solve systems of equations and helps in predicting the behavior of the system in question.

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Most popular questions from this chapter

Write each system in the form \(A X=B .\) Then solve the system by entering \(A\) and \(B\) into your graphing utility and computing \(A^{-1} B\). $$\left\\{\begin{aligned}v-3 x+z &=-3 \\\w+x+y &=-1 \\\x+w-x+4 y &=7 \\\v+w+x+y+z &=-8 \\\v+w+x+y+z &=8\end{aligned}\right.$$

Describe how to subtract matrices.

Low-resolution digital photographs use \(262,144\) pixels in a \(512 \times 512\) grid. If you enlarge a low-resolution digital photograph enough, describe what will happen.

Solve the system: $$\left\\{\begin{aligned}x^{2}-2 y^{2} &=-1 \\\2 x^{2}-y^{2} &=1\end{aligned}\right.$$ (Section 7.4, Example 3)

Will help you prepare for the material covered in the next section. Simplify the expression in each exercise. $$2(-5)-(-3)(4)$$
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