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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Adjoint Matrix">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Fri Mar 28 15:33:51 1997}
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\section{Adjoint Matrix\label{Adjoint Matrix}}

Consider the $n\times n$ matrix $A=\left[ 
\begin{array}{lllll}
a_{11} & a_{12} & a_{13} & ... & a_{1n} \\ 
a_{21} & a_{22} & a_{23} & ... & a_{2n} \\ 
. & . &  &  & . \\ 
. & . &  &  & . \\ 
. & . &  &  & . \\ 
a_{n1} & a_{n2} & a_{n3} & ... & a_{nn}
\end{array}
\right] $.

\subsubsection{Cofactor\label{cofactor}}

The cofactor of an entry $a_{ij}$ is defined to be the number $A_{ij}=\left(
-1\right) ^{i+j}D_{ij}$, where $D_{ij}$ is the determinant of the $\left(
n-1\right) \times \left( n-1\right) $ matrix obtained from $A$ by omitting
the $i^{th}$ row and $j^{th}$ column of $A$.

\subsubsection{\textbf{Adjoint Matrix\label{adjoint matrix}}}

For each entry $a_{ij}$ of $A$, let $A_{ij}$ be the cofactor of $a_{ij}$.
The adjoint matrix of $A$ is defined to be the $n\times n$ matrix whose
entry in the $i^{th}$ row and $j^{th}$ column is $A_{ji}$.

\begin{itemize}
\item  If $A$ is a nonsingular square matrix, then 
\[
A^{-1}=\frac{1}{\det A}\cdot \limfunc{adj}A
\]
\end{itemize}

\subsubsection{Cramer's Rule\label{cramer's rule}}

Let $A$ be a nonsingular $n\times n$ matrix and let $b$ be a vector in $%
\UNICODE{0x211d} ^{n}$. Then the components of the unique solution $x=\left(
x_{1},x_{2},...,x_{n}\right) $ of $Ax=b$ are 
\[
x_{i}=\frac{\det A_{i}}{\det A}\text{, \ \ }i=1,2,...,n
\]

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Cofactors and the adjugate (or classical adjoint) matrix}{%
}{}{../../Help/DM8-5.tex#Adjugate}}

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

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