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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Determinants and Cramer's Rule">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Determinants and Cramer's Rule\label{Determinants and Cramer's Rule}
}

If a matrix is \textbf{square} (that is, if it has the same number of rows
as columns), then you can assign to it a number called its \textbf{%
determinant}. Determinants can be used to solve matrix equations. They are
also useful in determining whether a matrix has an inverse, without actually
going through the process of trying to find its inverse.

\subsection{Determinant of a 2 x 2 Matrix}

Let $M$ be the $2\times 2$ matrix $\left[ 
\begin{array}{ll}
a & b \\ 
c & d
\end{array}
\right] $. Then the \textbf{determinant} of $M$, denoted $\left| M\right| $
or $\left| 
\begin{array}{ll}
a & b \\ 
c & d
\end{array}
\right| $, is given by 
\[
\left| M\right| =ad-bc
\]

\subsection{The Determinant of a Square Matrix\label%
{Definition of determinant of a square matrix}}

Now let $A$ be an $n\times n$ matrix.

\begin{enumerate}
\item  The \textbf{minor} $M_{ij}$ of the element $a_{ij}$ is the
determinant of the matrix obtained by deleting the $i$th row and the $j$th
column of $A$.

\item  The \textbf{cofactor} $A_{ij}$ of the element $a_{ij}$ is 
\[
A_{ij}=\left( -1\right) ^{i+j}M_{ij}
\]
\end{enumerate}

If $A$ is an $n\times n$ matrix, then the determinant of $A$ is obtained by
multiplying each element of the first row by its cofactor, and then adding
the results. In symbols, 
\[
\left| A\right| =\left| 
\begin{array}{llllll}
a_{11} & a_{12} & \cdot & \cdot & \cdot & a_{1n} \\ 
a_{21} & a_{22} & \cdot & \cdot & \cdot & a_{2n} \\ 
\cdot & \cdot & \cdot &  &  & \cdot \\ 
\cdot & \cdot &  & \cdot &  & \cdot \\ 
\cdot & \cdot &  &  & \cdot & \cdot \\ 
a_{n1} & a_{n2} & \cdot & \cdot & \cdot & a_{nn}
\end{array}
\right| =a_{11}A_{11}+a_{12}A_{12}+\cdot \cdot \cdot +a_{1n}A_{1n} 
\]

\subsection{Invertibility Criterion\label%
{Determinant criterion for invertibility}}

If $A$ is a square matrix, then $A$ has an inverse if and only if $\left|
A\right| \neq 0.$

\subsection{Row and Column Transformations of a Determinant\label%
{Effect of row and column transformations on a determinant}}

If $A$ is a square matrix, and if the matrix $B$ is obtained from $A$ by
adding a multiple of one row to another, or a multiple of one column to
another, then $\left| A\right| =\left| B\right| .$

\subsection{Cramer's Rule\label{Cramer's Rule}}

If a system of $n$ linear equations in the $n$ variables $%
a_{1},x_{2},...,x_{n}$ is equivalent to the matrix equation $DX=B$, and if $%
\left| D\right| \neq 0$, then its solutions are 
\[
x_{1}=\frac{\left| D_{x_{1}}\right| }{\left| D\right| }\text{,\quad }x_{2}=%
\frac{\left| D_{x_{2}}\right| }{\left| D\right| }\text{, }...\text{,\quad }%
x_{n}=\frac{\left| D_{x_{n}}\right| }{\left| D\right| }
\]
where $D_{x_{i}}$ is the matrix obtained by replacing the $i$th column of $D$
by the $n\times 1$ matrix $B$.

\subsection{Related Scientific Notebook Topics}

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

{\small \hyperref{Computing a determinant}{}{}{%
../../Help/DM8-5.tex#Determinant}}

{\small \hyperref{Determinant, definition by permutations}{}{}{%
../../Help/DM8-5.tex#Determinant}}

\noindent \hrulefill

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\noindent \hrulefill

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