\documentclass{article}
\title{Determinants and Solutions of Linear Systems of Equations}
\author{Megan Zwolinski}
\begin{document}
\maketitle

\tableofcontents

\section{Introduction}
In this paper, we will study determinants and solutions of linear systems of equations in some detail.  We will learn the basics for each and expand on them.  
\section{Determinants}
A determinant is a mathematical object which is very useful in the analysis and solution of systems of linear equations.  Determinants are only defined for square matrices.  A square matrix has horizontal and vertical dimensions that are the same (i.e., an $nxn$ matrix).  The difference between the form of a matrix and a determinant of a matrix is that a determinant is displayed using straight lines in-place of the square brackets. The determinant is a scalar quantity, which means a one-component quantity. The determinant is most often used to: 
\begin{itemize}
	\item test whether or not a matrix has an inverse
	\item test for linear dependence of vectors (in certain situations)
	\item test for existence/uniqueness of solutions of linear systems of equations
\end{itemize}

\section{An $nxn$ Matrix}  In an $nxn$ matrix, we follow certain rules for the appearance of the matrix.   
We let $v_i$ symbolize the $i^{th}$ row.  $(a_{i1},a_{i2},...,a_{in})$.  
If this row were to be multiplied by $\alpha$, the the row would appear to be $(\alpha a_{i1},\alpha a_{i2},...,\alpha a_{in})$.  
Also, if two rows were added, the $i^{th}$ row and the $j^{th}$ column, we would have $(a_{i1}+a_{j1},a_{i2}+a_{j2},...,a_{in}+a_{jn})$.  
A unit matrix appears with the following rows:  (1,0,0,...,0),(0,1,0,...,0),...,(0,0,0,...,1).  Also, the letters $e_1,...,e_n$ describe a unit row.  

In an $nxn$ matrix there are a few forms in which a determinant is recognized. First of all the determinant symbolizes the function of the $n^2$ variables $a_{ij}(i,j=1,2,...,n)$.  The determinant for this function can be written as:

\begin{displaymath}
D =
\left|
\begin{array}{cccc}
a_{11}&a_{12}&\dots&a_{1n}\\
a_{21}&a_{22}&\dots&a_{2n}\\
\vdots&      &\ddots&\vdots\\
a_{n1}&a_{n2}&\dots&a_{nn}\\
\end{array}
\right|
= |a_{ij}| = D(v_1,v_2,...v_n)
\end{displaymath}

\section{Properties of Determinants}
\begin{enumerate}
	\item $D(v_1,v_2,...,v_i,...,v_n)=D(v_1,v_2,...,v_i + v_j,...v_n)$  $(i \neq j)$ (\textbf{Invariance})
	\item $D(v_1,v_2,...,\alpha v_i,...,v_n)= \alpha D(v_1,v_2,...,v_i + v_j,...v_n)$  (\textbf{Homogeneity})
	\item $D(e_1,e_2,...,e_n)=1$ (\textbf{Normalization})
\end{enumerate}

\section{Rules for Determinants}  Determinants are either written as $|A|$ or $det A$.  
Now say that we delete the $i^{th}$ row and the $j^{th}$ column a $(n-1)x(n-1)$ submatrix $A_{ij}$ is formed.  
The determinant of this submatrix is the \textit{minor} element of $a_{ij}$.  
The \textit{cofactor} of $a_{ij}$ is $(-1)^{i+j}|A_{ij}|$.  We also write the cofactor as $A_{ij}^*$.  
The following are some rules for determinants.  

\textit{a.}		$|A| = |A'|, A' = transpose of A = (a_ji))$

\textit{b.} 	If two rows (or columns) of A are interchanged, producing a matrix $A_1$, then $|A|$ = $-|A|$

\textit{c.} 	If two rows (or columns) of A are identical, then |A|=0

\textit{d.} 	If a row (or column), $v$, of A is replaced by $kv$ producing a matrix $A_1$, then $|A|$=$k|A|$.

\textit{e.} 	If a scalar multiple $kv$, of the $ith$ row (or column) is added to the $jth$ row (or column) $v_j$, $(i \neq j)$ and the matrix $A_1$ results, then $|A|$ = $|A_1|$.

\textit{f.} 	A determinant may be evaluated in terms of cofactors: $|A|$ $=$ $$\sum_{i=1}^n a_{ij}A_{ij}^*     \   1 \leq j \leq n $$  $$=\sum_{i=1}^n a_{ij}A_{ij}^*    \        1 \leq i \leq n $$ 



\section{$2X2$ Matrix}  The most basic determinant is found using a $2x2$ matrix in the form $$A=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]$$
The determinant of a $2x2$ matrix is found using the following formula:  $$\left|A\right|=det(A)=\left|\begin{array}{cc}a&b\\c&d\end{array}\right|=ad-bc$$

\section{Example 1}
\textbf{$2x2$ Matrix}  Using the matrix $$A=\left[\begin{array}{cc}4&5\\2&3\end{array}\right]$$  the determinant would be $$\left|A\right|=det(A)=\left|\begin{array}{cc}4&5\\2&3\end{array}\right|=4*3-5*2=2$$

\section{$3x3$ Matrix} The next kind of matrix studied is the $3x3$ matrix.   The form of this matrix is $$B=\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right]$$
The determinant of a $3x3$ matrix is found using the following formula:  $$\left|B\right|=det(B)=\left|\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right|=a\left|\begin{array}{cc}e&f\\h&i\end{array}\right|-b\left|\begin{array}{cc}d&f\\g&i\end{array}\right|+c\left|\begin{array}{cc}d&e\\g&h\end{array}\right|$$
\section{Example 2}  
\textbf{$3x3$ Matrix}  Using the matrix $$B=\left[\begin{array}{ccc}1&3&5\\2&4&6\\0&2&4\end{array}\right]$$ the determinant would be $$\left|B\right|=det(B)=\left|\begin{array}{ccc}1&3&5\\2&4&6\\0&2&4\end{array}\right|=1\left|\begin{array}{cc}4&6\\2&4\end{array}\right|-3\left|\begin{array}{cc}2&6\\0&4\end{array}\right|+5\left|\begin{array}{cc}2&4\\0&2\end{array}\right|$$



\section{Solution of Linear Systems of Equations} Consider the system of $n$ linear equations in $n$ unknowns $x_1,x_2,...,x_n$ 
$$\sum_{j=1}^n a_{ij}x_j=b_i     \   (i=1,2,...,n)$$ 

\section{Cramer's Rule} The above formula is Cramer's Rule.  It states that if $|A|=|a_{ij}|\neq0$, then 
$$\sum_{j=1}^n a_{ij}x_j=b_i     \   (i=1,2,...,n)$$ possesses a unique solution given by 
$$x_r = \frac{\sum_{i=1}^n A_{ir}^*b_i}{|A|}     \   (r=1,2,...,n)$$ 
Cramer's rule is used to solve a set of $n$ linear equations in $n$ unknowns. It uses determinants to obtain the solution. 

\section{The Alternative Theorem} This formula that results from Cramer's Rule is the Alternative Theorem.  Also described as the homogeneous system 
$$\sum_{j=1}^n a_{ij}x_j=0     \   (i=1,2,...,n)$$ 
possesses a non-trivial solution (i.e., a solution other than $x_1=x_2=...=x_n=0$) 
if and only if $|A|$=0.  If for a fixed  
$A=(a_{ij})$ there are solutions to the non-homogeneous system 
$$\sum_{j=1}^na_{ij}x_j=b_i     \   (i=1,2,...,n)$$ 
for every selection of the quantities $b_i$, then $|A|\neq0$ and the homogeneous system had only the trivial solution.

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