Elementary row transformations. If we want to perform an elementary row transformation on a matrix A, it is enough to pre-multiply A by the elemen-tary matrix obtained from the identity by the same transformation. This is illustrated below for each of the three elementary row transformations. 1.5.2 Elementary Matrices and Elementary Row Opera-tions.
Preface
Preliminaries
Sets And Set Notation
Functions
The Number Line And Algebra Of The Real Numbers
Ordered fields
The Complex Numbers
The Fundamental Theorem Of Algebra
Exercises
Completeness of R
Well Ordering And Archimedean Property
Division
Systems Of Equations
Exercises
Fn
Algebra in Fn
Exercises
The Inner Product In Fn
What Is Linear Algebra?
Exercises
Linear Transformations
Matrices
Exercises
Linear Transformations
Some Geometrically Dened Linear Transformations
The Null Space Of A Linear Transformation
Subspaces And Spans
An Application To Matrices
Matrices And Calculus
Exercises
Determinants
Basic Techniques And Properties
Exercises
The Mathematical Theory Of Determinants
The Cayley Hamilton Theorem
Block Multiplication Of Matrices
Exercises
Row Operations
Elementary Matrices
The Rank Of A Matrix
The Row Reduced Echelon Form
Rank And Existence Of Solutions To Linear Systems
Fredholm Alternative
Exercises
Some Factorizations
LU Factorization
Finding An LU Factorization
Solving Linear Systems Using An LU Factorization
The PLU Factorization
Justification For The Multiplier Method
Existence For The PLU Factorization
The QR Factorization
Exercises
Spectral Theory
Eigenvalues And Eigenvectors Of A Matrix
Some Applications Of Eigenvalues And Eigenvectors
Exercises
Schur’s Theorem
Trace And Determinant
Quadratic Forms
Second Derivative Test
The Estimation Of Eigenvalues
Advanced Theorems
Exercises
Cauchy’s Interlacing Theorem for Eigenvalues
Learning Outcomes
Write the augmented matrix for a system of equations.
Perform row operations on an augmented matrix.
A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.
For example, consider the following [latex]2times 2[/latex] system of equations.
Notice that the matrix is written so that the variables line up in their own columns: x-terms go in the first column, y-terms in the second column, and z-terms in the third column. It is very important that each equation is written in standard form [latex]ax+by+cz=d[/latex] so that the variables line up. When there is a missing variable term in an equation, the coefficient is 0.
How To: Given a system of equations, write an augmented matrix
Write the coefficients of the x-terms as the numbers down the first column.
Write the coefficients of the y-terms as the numbers down the second column.
If there are z-terms, write the coefficients as the numbers down the third column.
Draw a vertical line and write the constants to the right of the line.
Example: Writing the Augmented Matrix for a System of Equations
Write the augmented matrix for the given system of equations.
Writing a System of Equations from an Augmented Matrix
We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the system of equations in standard form.
Example: Writing a System of Equations from an Augmented Matrix Form
Find the system of equations from the augmented matrix.
Now that we can write systems of equations in augmented matrix form, we will examine the various row operations that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows.
Performing row operations on a matrix is the method we use for solving a system of equations. In order to solve the system of equations, we want to convert the matrix to row-echelon form, in which there are ones down the main diagonal from the upper left corner to the lower right corner and zeros in every position below the main diagonal as shown.
[latex]begin{array}{c}text{Row-echelon form} left[begin{array}{ccc}1& a& b 0& 1& d 0& 0& 1end{array}right]end{array}[/latex]
We use row operations corresponding to equation operations to obtain a new matrix that is row-equivalent in a simpler form. Here are the guidelines to obtaining row-echelon form.
In any nonzero row, the first nonzero number is a 1. It is called a leading 1.
Any all-zero rows are placed at the bottom of the matrix.
Any leading 1 is below and to the right of a previous leading 1.
Any column containing a leading 1 has zeros in all other positions in the column.
To solve a system of equations we can perform the following row operations to convert the coefficient matrix to row-echelon form and do back-substitution to find the solution.
Multiply a row by a constant. (Notation: [latex]c{R}_{i}[/latex] )
Add the product of a row multiplied by a constant to another row. (Notation: [latex]{R}_{i}+c{R}_{j}[/latex])
Each of the row operations corresponds to the operations we have already learned to solve systems of equations in three variables. With these operations, there are some key moves that will quickly achieve the goal of writing a matrix in row-echelon form. To obtain a matrix in row-echelon form for finding solutions, we use Gaussian elimination, a method that uses row operations to obtain a 1 as the first entry so that row 1 can be used to convert the remaining rows.
A General Note: Gaussian Elimination
The Gaussian elimination method refers to a strategy used to obtain the row-echelon form of a matrix. The goal is to write matrix [latex]A[/latex] with the number 1 as the entry down the main diagonal and have all zeros below.
The first row already has a 1 in row 1, column 1. The next step is to multiply row 1 by [latex]-2[/latex] and add it to row 2. Then replace row 2 with the result.