%% This document created by Scientific Notebook (R) Version 3.0


\documentclass[12pt,thmsa]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{sw20jart}

%TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart}

%TCIDATA{<META NAME="ViewSettings" CONTENT="23">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Properties of Determinants">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Fri Mar 28 15:33:54 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=Reference.cst}
%TCIDATA{PageSetup=72,72,72,72,0}
%TCIDATA{<LINK REL="contents" HREF="..\Reference\ReferenceIndex.tex#Reference Index">}
%TCIDATA{<LINK REL="next" HREF="MA04_02.tex">}
%TCIDATA{<LINK REL="parent" HREF="MA_contents.tex">}
%TCIDATA{<LINK REL="previous" HREF="MA03_04.tex">}
%TCIDATA{AllPages=
%F=36,\PARA{038<p type="texpara" tag="Body Text" >\hfill \thepage}
%}


\input{tcilatex}
\begin{document}


\section{ Properties of the Determinant\label{Properties of the Determinant}}

\subsubsection{Determinant\label{determinant}}

The determinant function associates with every square matrix $A$ a unique
number (denoted $\det A$) called the \textbf{determinant} of $A$. Some of
the main uses of the determinant are the following:

\begin{enumerate}
\item  The determinant gives a test for invertibility. If $\det A\neq 0$,
then $A$ is invertible.

\item  The determinant of $A$ equals the volume of the parallelopiped $P$ in 
$n$-dimensional space that is formed from the rows of $A$.
\end{enumerate}

\begin{definition}
The determinant of a $2\times 2$ matrix $\left[ 
\begin{array}{cc}
a & b \\ 
c & d
\end{array}
\right] $ is defined by the formula $ad-bc$.
\end{definition}

\subsubsection{Properties Of the Determinant\label%
{properties of the determinant}}

Let $A$ and $B$ be $n\times n$ matrices. Then

\begin{enumerate}
\item  If a scalar multiple of one row of $A$ is added to another row of $A$%
, then the determinant of the resulting matrix is equal to $\det A$.

\item  If any two rows of $A$ are interchanged, then the determinant of the
resulting matrix is equal to $-\det A$.

\item  If any row of a $A$ is multiplied by a scalar $c$, then the
determinant of the resulting matrix is $c\det A$.

\item  The determinant of the $n\times n$ matrix $I$ is equal to $1$.

\item  If $A$ has a zero row, then its determinant is equal to zero.

\item  If $A$ has two identical rows, then its determinant is equal to zero.

\item  If $A$ is upper triangular, then $\det A$ is equal to the product of
the diagonal entries of $A$.

\item  $\det A=\det A^{T}$

\item  $\det \left( AB\right) =\left( \det A\right) \left( \det B\right)
\bigskip $
\end{enumerate}

\begin{theorem}
A square matrix $A$ is \hyperref{nonsingular}{}{}{MA01_06.tex#nonsingular}
(invertible) if and only if $\det A\neq 0$.$\bigskip $
\end{theorem}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Determinant}{}{}{../../Help/DM8-5.tex#Determinant}}

\noindent \hrulefill

\begin{center}
\begin{tabular}{ccc}
{\small \hyperref{\FRAME{itbpF}{13.9375pt}{14.4375pt}{3pt}{}{}{nblogo.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 13.9375pt;height 14.4375pt;depth
3pt;original-width 24.0625pt;original-height 24.8125pt;cropleft "0";croptop
"0.9959";cropright "0.9937";cropbottom "0";filename
'Nblogo.wmf';file-properties "XNPEU";}}Computing Techniques}{}{}{%
../../Help/DMindex.tex}} & {\small \hyperref{\FRAME{itbpF}{13.9375pt}{%
14.4375pt}{3pt}{}{}{nblogo.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
13.9375pt;height 14.4375pt;depth 3pt;original-width
24.0625pt;original-height 24.8125pt;cropleft "0";croptop "0.9959";cropright
"0.9937";cropbottom "0";filename 'Nblogo.wmf';file-properties "XNPEU";}}%
General Information}{}{}{../../Help/Index.tex}} & {\small \hyperref{\FRAME{%
itbpF}{13.9375pt}{14.4375pt}{3pt}{}{}{nblogo.wmf}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"PICT";valid_file "F";width 13.9375pt;height 14.4375pt;depth
3pt;original-width 24.0625pt;original-height 24.8125pt;cropleft "0";croptop
"0.9959";cropright "0.9937";cropbottom "0";filename
'Nblogo.wmf';file-properties "XNPEU";}}Reference Library}{}{}{%
../ReferenceIndex.tex}}
\end{tabular}
\end{center}

\noindent \hrulefill

\end{document}
