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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Inconsistent and Dependent Systems">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Inconsistent and Dependent Systems\label%
{Inconsistent and Dependent Systems}}

\vspace{1pt}

\subsection{\protect\vspace{1pt}Echelon Form of a Matrix\label%
{Echelon Form of a Matrix}}

A matrix is in \textbf{echelon form} if it has the following properties:

\begin{enumerate}
\item  The first nonzero number in each row (reading from left to right) is
1. This is called the\emph{\ leading entry} of the row.

\item  The leading entry in each row is to the right of the leading entry in
the row immediately above it.

\item  Rows that consist entirely of $0$'s are at the bottom of the matrix.
\end{enumerate}

\subsection{Solving a System in Echelon Form\label%
{Solving a System in Echelon Form}}

Suppose the matrix that represents a system of linear equations has been
transformed by Gaussian elimination into echelon form.

\begin{enumerate}
\item  If the echelon form contains a row that represents the equation $0=c$%
, where $c$ is nonzero, then the system has\emph{\ no solution.}

\item  If each of the variables in the echelon form of the system is a
leading variable, then the system has \emph{exactly one solution}, which you
find using back substitution.

\item  If not all of the variables in the echelon form are leading
variables, then the system has \emph{infinitely many solutions}. To solve
the system, use back substitution to express the leading variables in terms
of the arbitrary values.
\end{enumerate}

\subsection{Reduced Echelon Form of a Matrix}

A matrix is in \textbf{reduced echelon form} if it is in echelon form and
every number above and below each leading entry is $0$.

\subsection{Related Reference Topics\hfill }

{\small \hyperref{Solving Systems -- Forward Elimination, Gaussian
Elimination}{}{}{MA01_03.tex}}

{\small \hyperref{Matrix Solution of Systems}{}{}{MA01_05.tex}}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Matrix Algebra}{}{}{../../Help/DM8.tex}}

{\small \hyperref{Row Operations and Echelon Forms}{}{}{%
../../Help/DM8-3.tex#Matrices, row operations}}

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