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\begin{document}


\section{Row Operations \label{Matrices, row operations}and Echelon Forms}

One of the elementary 
\index{Matrices@Matrices!echelon forms@echelon forms}\label%
{Matrices, echelon forms} applications of matrix arrays is 
\index{Matrices@Matrices!row operations@row operations}storing and
manipulating coefficients of systems of linear equations. The various steps
that you carry out in applying the technique of elimination to a system of
linear equations \ 
\begin{eqnarray*}
a_{11}x_{1}+a_{12}x_{2}+\ldots +a_{1n}x_{n} &=&b_{1} \\
a_{21}x_{1}+a_{22}x_{2}+\ldots +a_{2n}x_{n} &=&b_{2} \\
\vdots \qquad \qquad \qquad &\vdots &\;\vdots \\
a_{m1}x_{1}+a_{m2}x_{2}+\ldots +a_{mn}x_{n} &=&b_{m}
\end{eqnarray*}
can be applied equally well to the matrix of coefficients and scalars 
\[
\left[ 
\begin{array}{ccccc}
a_{11} & a_{12} & \ldots & a_{1n} & b_{1} \\ 
a_{21} & a_{22} & \ldots & a_{2n} & b_{2} \\ 
\vdots & \vdots & \ldots & \vdots & \vdots \\ 
a_{m1} & a_{m2} & \ldots & a_{mn} & b_{m}
\end{array}
\right] 
\]
For this and numerous other reasons, you do \textsl{elementary row operations%
} on matrices. The goal of elementary row operations is to put the matrix in
a special form, such as a \textsl{row echelon form,} where the number of
leading zeroes increases as the row number increases. The system provides
several choices for obtaining a row echelon form 
\index{Row echelon forms@Row echelon forms}\label{Row echelon forms}, one of
which gives the \textsl{reduced row echelon form} satisfying the following
conditions.

\begin{itemize}
\item  The number of leading zeroes increases as the row number increases.

\item  The first nonzero entry in each nonzero row is equal to $1$.

\item  Each column that contains the leading nonzero entry for any row
contains only zeroes above and below that entry.
\end{itemize}

\subsection{\ Gaussian Elimination and Row Echelon Form\label%
{Matrices, fraction-free Gaussian elimination}
\index{Matrices@Matrices!Gaussian elimination@Gaussian elimination}\label%
{Matrices, Gaussian elimination}}

The three row echelon forms that can be obtained directly are illustrated in
the following examples. 
\index{Matrices@Matrices!fraction-free Gaussian elimination@fraction-free Gaussian elimination}

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Fraction-free Gaussian Elimination}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{cc}
a & b \\ 
c & d
\end{array}
\right] $, fraction-free Gaussian elimination: $\left[ 
\begin{array}{cc}
a & b \\ 
0 & da-bc
\end{array}
\right] $

\ 

$\left[ 
\begin{array}{rrr}
8 & 2 & 3 \\ 
2 & -5 & 8
\end{array}
\right] $, fraction-free Gaussian elimination: $\left[ 
\begin{array}{rrr}
8 & 2 & 3 \\ 
0 & -44 & 58
\end{array}
\right] \smallskip $
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Reduced Row Echelon Form}\label%
{Matrices, reduced row echelon form}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{cc}
a & b \\ 
c & d
\end{array}
\right] $, 
\index{Matrices@Matrices!reduced row echelon form@reduced row echelon form}%
Reduced row echelon form: $\left[ 
\begin{array}{cc}
1 & 0 \\ 
0 & 1
\end{array}
\right] $

\ 

$\left[ 
\begin{array}{rrr}
8 & 2 & 3 \\ 
2 & -5 & 8
\end{array}
\right] $, Reduced row echelon form: $\left[ 
\begin{array}{ccr}
1 & 0 & 
\frac{31}{44}\vspace{8pt} \\ 
0 & 1 & -\frac{29}{22}
\end{array}
\right] \smallskip $
\end{quotation}

\strut

\subsection{\ Elementary Row Operations\label%
{Matrices, elementary row operations}\label{Elementary matrices}}

You can do 
\index{Matrices@Matrices!elementary row operations@elementary row operations}%
elementary row operations 
\index{Elementary row operations@Elementary row operations}by multiplying on
the left by appropriate \textsl{elementary matrices}, the matrices obtained
from an identity matrix by applying the same elementary row operation as you
want to perform on your matrix. The technique is illustrated in the
following examples.

To create an elementary matrix, choose \textsf{Fill Matrix} from the \textsf{%
Matrices} submenu; get an identity matrix of the appropriate dimension by
making choices in the \textsf{Fill Matrix} dialog box, and perform an
elementary row operation by editing the identity matrix. Choose \textsf{%
Evaluate} to get the following products.

\begin{quote}
$\blacktriangleright $ Add $\lambda $ times row $3$ to row $1$
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{ccc}
1 & 0 & \lambda \\ 
0 & 1 & 0 \\ 
0 & 0 & 1
\end{array}
\right] \left[ 
\begin{array}{rrr}
-5 & -2 & -1 \\ 
3 & -6 & 2 \\ 
1 & 4 & 1
\end{array}
\right] =\,\left[ 
\begin{array}{ccc}
-5+\lambda & -2+4\lambda & -1+\lambda \\ 
3 & -6 & 2 \\ 
1 & 4 & 1
\end{array}
\right] \smallskip $
\end{quotation}

\begin{quote}
$\blacktriangleright $ Interchange rows $2$ and $3$
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{ccc}
1 & 0 & 0 \\ 
0 & 0 & 1 \\ 
0 & 1 & 0
\end{array}
\right] \left[ 
\begin{array}{rrr}
-50 & -12 & -18 \\ 
31 & -26 & -62 \\ 
1 & -47 & -91
\end{array}
\right] =\,\left[ 
\begin{array}{rrr}
-50 & -12 & -18 \\ 
1 & -47 & -91 \\ 
31 & -26 & -62
\end{array}
\right] \smallskip $
\end{quotation}

\begin{quote}
$\blacktriangleright $ Multiply row $2$ by $\lambda $
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{ccc}
1 & 0 & 0 \\ 
0 & \lambda & 0 \\ 
0 & 0 & 1
\end{array}
\right] \left[ 
\begin{array}{rrr}
80 & -2 & -18 \\ 
33 & -26 & 82 \\ 
14 & -47 & -91
\end{array}
\right] =\,\left[ 
\begin{array}{lll}
80 & -2 & -18 \\ 
33\lambda & -26\lambda & \;\;82\lambda \\ 
14 & -47 & -91
\end{array}
\right] \smallskip $
\end{quotation}

\strut

\subsection{\ Related topics}

\begin{itemize}
\item  
\hyperref{Working With Matrices}{}{}{DM8-1.tex#Matrix Algebra}

\item  \hyperref{Standard Operations}{}{}{DM8-2.tex#Standard matrix
operations}

\item  \hyperref{Systems of Linear Equations}{}{}{DM8-4.tex#Systems of
equations}

\item  \hyperref{Matrix Equations}{}{}{DM8-4.tex#Matrix equations}

\item  \hyperref{Matrix Operators}{}{}{DM8-5.tex#Trace}

\item  \hyperref{Polynomials and Vectors Associated with a Matrix}{}{}{%
DM8-6.tex#Characteristic, polynomial}

\item  \hyperref{Vector Spaces Associated with a Matrix}{}{}{DM8-7.tex#Row
space}

\item  \hyperref{Special Forms of Matrices}{}{}{DM8-8.tex#Smith normal form}

\item  \hyperref{Exercises and Solutions}{}{}{DM8-9.tex#Exercises}
\end{itemize}

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