## Sunday, July 20, 2014

### Inverse of a 3x3 Matrix

In my previous post
(http://mikemstech.blogspot.com/2014/07/c-matrix-inversion-with-latex-output.html)
I demonstrated an application that can generate the steps to show the
inversion of a matrix by Gauss Jordan elimination.

In a few posts, I plan to answer the following questions:
What is the inverse of a 1x1 Matrix?
What is the inverse of a 2x2 Matrix?
What is the inverse of a 3x3 Matrix?
What is the inverse of a 4x4 Matrix?

Back to Mike's Big Data, Data Mining, and Analytics Tutorial

The inverse of a 3x3 matrix is defined as follows. For a 3x3 matrix:

$$A = \begin{pmatrix}a & b & c \\ d & e & f \\ g & h & i\end{pmatrix}$$

$$A^{-1} = \begin{pmatrix} \frac{f h-e i}{c e g-b f g-c d h+a f h+b d i-a e i} & \frac{c h - b i}{- c e g + b f g + c d h - a f h - b d i + a e i} & \frac{c e - b f}{c e g - b f g - c d h + a f g + b d i - a e i } \\ \frac{f g - d i}{- c e g + b f g + c d h - a f h - b d i + a e i} & \frac{c g - a i}{c e g - b f g - c d h+ a f h + b d i - a e i} & \frac{c d - a f}{- c e g + b f g + c d h - a f h - b d i + a e i} \\ \frac{e g - d h}{c e g - b f g - c d h + a f h + b d i - a e i} & \frac{b g - a h}{- c e g + b f g + c d h - a f h - b d i + a e i} & \frac{b d - a e}{c e g - b f g - c d h + a f h + b d i - a e i} \end {pmatrix}$$

The latex code generated for a 3x3 inverse is the following:

\documentclass{article}

% This is the output from LatexMatrixInverse 1.0 for a matrix with rank 3
% http://mikemstech.blogspot.com

\usepackage{geometry}

% Note: you should probably use pdflatex to compiile this file.
% Other processors are known to have some issues with using
% 'geometry' to set paper size

% Adjust the page size here if output is wrapping in a bad way.
% Default is 8.5 x 11 in (Letter)
\geometry{papersize={40in,14in}}

%Import AMS Latex packages
\usepackage{amsmath, amssymb}
\setcounter{MaxMatrixCols}{7}

%Variable definition

\begin{document}
% Definition of initial A
% A row 1
\newcommand{\ARbCb}{a}
\newcommand{\ARbCc}{b}
\newcommand{\ARbCd}{c}

% A row 2
\newcommand{\ARcCb}{d}
\newcommand{\ARcCc}{e}
\newcommand{\ARcCd}{f}

% A row 3
\newcommand{\ARdCb}{g}
\newcommand{\ARdCc}{h}
\newcommand{\ARdCd}{i}

% Definition of initial B
\newcommand{\BRb}{j}
\newcommand{\BRc}{k}
\newcommand{\BRd}{l}

LatexMatrixInverse 1.0 Output for rank 3, ShowIntermediateSteps is True.

Given the following initial matrices:
\begin{equation*}
A = \begin{pmatrix}\ARbCb
&\ARbCc
&\ARbCd
\\\ARcCb
&\ARcCc
&\ARcCd
\\\ARdCb
&\ARdCc
&\ARdCd
\end{pmatrix}B = \begin{pmatrix}\BRb\\ \BRc\\ \BRd\end{pmatrix}\end{equation*}
We want to find $A^{-1}$ and $A^{-1} B$...
\begin{equation*}
\left ( \begin{array}{ccc|ccc|c}\ARbCb
&\ARbCc
&\ARbCd
&1
&0
&0
&\BRb\\
\ARcCb
&\ARcCc
&\ARcCd
&0
&1
&0
&\BRc\\
\ARdCb
&\ARdCc
&\ARdCd
&0
&0
&1
&\BRd\end{array} \right )
\end{equation*}
\begin{equation*}
\xrightarrow{\frac{1}{\ARbCb} R_{1}}
\left ( \begin{array}{ccc|ccc|c}1
&\frac{\ARbCc}{\ARbCb}
&\frac{\ARbCd}{\ARbCb}
&\frac{1}{\ARbCb}
&0
&0
&\frac{\BRb}{\ARbCb}\\
\ARcCb
&\ARcCc
&\ARcCd
&0
&1
&0
&\BRc\\
\ARdCb
&\ARdCc
&\ARdCd
&0
&0
&1
&\BRd\end{array} \right )
\end{equation*}
\begin{equation*}
\xrightarrow{R_{2} - \left ( \ARcCb\right ) R_{1}}
\left ( \begin{array}{ccc|ccc|c}1
&\frac{\ARbCc}{\ARbCb}
&\frac{\ARbCd}{\ARbCb}
&\frac{1}{\ARbCb}
&0
&0
&\frac{\BRb}{\ARbCb}\\
0
&\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
&\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )
&0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )
&1
&0
&\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )\\
\ARdCb
&\ARdCc
&\ARdCd
&0
&0
&1
&\BRd\end{array} \right )
\end{equation*}
\begin{equation*}
\xrightarrow{R_{3} - \left ( \ARdCb\right ) R_{1}}
\left ( \begin{array}{ccc|ccc|c}1
&\frac{\ARbCc}{\ARbCb}
&\frac{\ARbCd}{\ARbCb}
&\frac{1}{\ARbCb}
&0
&0
&\frac{\BRb}{\ARbCb}\\
0
&\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
&\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )
&0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )
&1
&0
&\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )\\
0
&\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
&\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )
&0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right )
&0
&1
&\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )\end{array} \right )
\end{equation*}
\begin{equation*}
\xrightarrow{\frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} R_{2}}
\left ( \begin{array}{ccc|ccc|c}1
&\frac{\ARbCc}{\ARbCb}
&\frac{\ARbCd}{\ARbCb}
&\frac{1}{\ARbCb}
&0
&0
&\frac{\BRb}{\ARbCb}\\
0
&1
&\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
&\frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
&\frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
&0
&\frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\\
0
&\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
&\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )
&0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right )
&0
&1
&\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )\end{array} \right )
\end{equation*}
\begin{equation*}
\xrightarrow{R_{3} - \left ( \ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )\right ) R_{2}}
\left ( \begin{array}{ccc|ccc|c}1
&\frac{\ARbCc}{\ARbCb}
&\frac{\ARbCd}{\ARbCb}
&\frac{1}{\ARbCb}
&0
&0
&\frac{\BRb}{\ARbCb}\\
0
&1
&\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
&\frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
&\frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
&0
&\frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\\
0
&0
&\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb
\right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left
( \frac{\ARbCc}{\ARbCb}\right )}\right )
&0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb
\right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{0 - \left (\ARcCb \right )
\left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )
&0 - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left (
\frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )
&1
&\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb
\right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\BRc - \left (\ARcCb \right )
\left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
\right )\end{array} \right )
\end{equation*}
\begin{equation*}
\xrightarrow{\frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc
- \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb
\right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )}\right )} R_{3}}
\left ( \begin{array}{ccc|ccc|c}1
&\frac{\ARbCc}{\ARbCb}
&\frac{\ARbCd}{\ARbCb}
&\frac{1}{\ARbCb}
&0
&0
&\frac{\BRb}{\ARbCb}\\
0
&1
&\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right )}
&\frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right )}
&\frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
&0
&\frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right )}\\
0
&0
&1
&\frac{0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc
- \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )
- \left (\ARdCc - \left (\ARdCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}
&\frac{0 - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{1}{\ARcCc -
\left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}
{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd
- \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}
&\frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}
{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}
&\frac{\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc
- \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )
- \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right )
\left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\end{array} \right )
\end{equation*}
\begin{equation*}
\xrightarrow{R_{2} - \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb
\right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right ) R_{3}}
\left ( \begin{array}{ccc|ccc|c}1
&\frac{\ARbCc}{\ARbCb}
&\frac{\ARbCd}{\ARbCb}
&\frac{1}{\ARbCb}
&0
&0
&\frac{\BRb}{\ARbCb}\\
0
&1
&0
&\frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
- \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
\right ) \linebreak \left ( \frac{0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb
\right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc -
\left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )
&\frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left ( \frac{0 - \left (\ARdCc - \left (\ARdCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd
- \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
\right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )}\right )}\right )
&0 - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )} \right ) \linebreak \left ( \frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb
\right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc
- \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )
&\frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
- \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )} \right ) \linebreak \left ( \frac{\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb
\right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc -
\left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc -
\left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )\\
0
&0
&1
&\frac{0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right )
\left ( \frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd
- \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left (
\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}
&\frac{0 - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{1}{\ARcCc - \left (\ARcCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb
\right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}
&\frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right )}\right )}
&\frac{\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
\right ) \left ( \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )}\right )}\end{array} \right )
\end{equation*}
\begin{equation*}
\xrightarrow{R_{1} - \left ( \frac{\ARbCd}{\ARbCb}\right ) R_{3}}
\left ( \begin{array}{ccc|ccc|c}1
&\frac{\ARbCc}{\ARbCb}
&0
&\frac{1}{\ARbCb} - \left (\frac{\ARbCd}{\ARbCb} \right ) \linebreak \left ( \frac{0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right ) -
\left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}
\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left (
\frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )
&0 - \left (\frac{\ARbCd}{\ARbCb} \right ) \linebreak \left ( \frac{0 - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
\right ) \left ( \frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left (
\frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd
- \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )
&0 - \left (\frac{\ARbCd}{\ARbCb} \right ) \linebreak \left ( \frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )
- \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left (
\frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )
&\frac{\BRb}{\ARbCb} - \left (\frac{\ARbCd}{\ARbCb} \right ) \linebreak \left ( \frac{\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )
- \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\BRc - \left (\ARcCb \right )
\left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right )
\left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd
- \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )\\
0
&1
&0
&\frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} -
\left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )} \right ) \linebreak \left ( \frac{0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb
\right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc -
\left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )
&\frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left (
\frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left ( \frac{0
- \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{1}{\ARcCc - \left (\ARcCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc -
\left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )
&0 - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left ( \frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )
- \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right )
\left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )
&\frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )} - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left ( \frac{\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right ) -
\left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\BRc - \left (\ARcCb \right )
\left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd -
\left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
\right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )\\
0
&0
&1
&\frac{0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right ) \right ) \left ( \frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc -
\left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}
&\frac{0 - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{1}{\ARcCc - \left (\ARcCb
\right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc
- \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}
&\frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc
- \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}
&\frac{\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
\right ) \left ( \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc -
\left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\end{array} \right )
\end{equation*}
\begin{equation*}
\xrightarrow{R_{1} - \left ( \frac{\ARbCc}{\ARbCb}\right ) R_{2}}
\left ( \begin{array}{ccc|ccc|c}1
&0
&0
&\frac{1}{\ARbCb} - \left (\frac{\ARbCd}{\ARbCb} \right ) \linebreak \left ( \frac{0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right )
- \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{0 - \left (\ARcCb \right ) \left (
\frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right )
\left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left (
\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )}\right )}\right ) - \left (\frac{\ARbCc}{\ARbCb} \right ) \linebreak \left ( \frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left (
\frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left ( \frac{0
- \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
\right ) \left ( \frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc -
\left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )\right )
&0 - \left (\frac{\ARbCd}{\ARbCb} \right ) \linebreak \left ( \frac{0 - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right ) \right ) \left ( \frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb
\right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right )
\left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )}\right )}\right ) - \left (\frac{\ARbCc}{\ARbCb} \right ) \linebreak \left ( \frac{1}{\ARcCc - \left (\ARcCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right )} - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb
\right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left ( \frac{0 - \left (\ARdCc - \left (\ARdCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd
- \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
\right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right )}\right )}\right )\right )
&0 - \left (\frac{\ARbCd}{\ARbCb} \right ) \linebreak \left ( \frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )
- \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left (
\frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right ) - \left (\frac{\ARbCc}{\ARbCb}
\right ) \linebreak \left ( 0 - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left ( \frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )
- \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left (
\frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )\right )
&\frac{\BRb}{\ARbCb} - \left (\frac{\ARbCd}{\ARbCb} \right ) \linebreak \left ( \frac{\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )
- \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )
- \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right )
\left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right ) - \left (\frac{\ARbCc}{\ARbCb}
\right ) \linebreak \left ( \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
- \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} \right )
\linebreak \left ( \frac{\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
\right ) \left ( \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
\right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
\right )}\right )\right )\\
0
&1
&0
&\frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} - \left (\frac{\ARcCd
- \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left (
\frac{0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left (
\frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd -
\left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left (
\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )
&\frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left ( \frac{0 - \left (\ARdCc - \left (\ARdCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd -
\left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left (
\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )
&0 - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
\right ) \linebreak \left ( \frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )
&\frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} - \left (
\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} \right )
\linebreak \left ( \frac{\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right ) \right ) \left ( \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right )}\right )}\right )\\
0
&0
&1
&\frac{0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left (
\frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd -
\left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left (
\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}
&\frac{0 - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{1}{\ARcCc - \left (\ARcCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}
&\frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
\right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )}\right )}
&\frac{\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right )
\left ( \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd
- \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left (
\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\end{array}
\right )
\end{equation*}
\begin{equation*}
A^{-1} = \begin{pmatrix}\frac{1}{\ARbCb} - \left (\frac{\ARbCd}{\ARbCb} \right ) \linebreak \left ( \frac{0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right )
- \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc
- \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc -
\left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc -
\left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right ) - \left (\frac{\ARbCc}{\ARbCb} \right ) \linebreak \left ( \frac{0 - \left (\ARcCb \right )
\left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left (
\frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left ( \frac{0 - \left (\ARdCb \right )
\left ( \frac{1}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{0 - \left (\ARcCb \right )
\left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left (
\frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right )
\left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )\right )
&
0 - \left (\frac{\ARbCd}{\ARbCb} \right ) \linebreak \left ( \frac{0 - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left (
\frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) -
\left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right ) - \left (\frac{\ARbCc}{\ARbCb} \right ) \linebreak \left (
\frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left ( \frac{0 - \left (\ARdCc - \left (\ARdCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd -
\left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right )
\left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
\right )}\right )\right )
&
0 - \left (\frac{\ARbCd}{\ARbCb} \right ) \linebreak \left ( \frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc
- \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}
{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right ) - \left (\frac{\ARbCc}{\ARbCb} \right ) \linebreak \left ( 0 -
\left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
\right ) \linebreak \left ( \frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )\right )
\\\frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} - \left (
\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} \right )
\linebreak \left ( \frac{0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
\right ) \left ( \frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd
- \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd
- \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )
&
\frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left ( \frac{0 - \left (\ARdCc - \left (\ARdCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{1}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd -
\left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right )
\left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
\right )}\right )
&
0 - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
\right ) \linebreak \left ( \frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right ) \right
) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}
\right )}\right )
\\\frac{0 - \left (\ARdCb \right ) \left ( \frac{1}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right )
\left ( \frac{0 - \left (\ARcCb \right ) \left ( \frac{1}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd
- \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left (
\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}
&
\frac{0 - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{1}{\ARcCc - \left (\ARcCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb
\right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}
&
\frac{1}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )
\right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}
\right )}\right )}
\end{pmatrix}A^{-1}B = \begin{pmatrix}\frac{\BRb}{\ARbCb} - \left (\frac{\ARbCd}{\ARbCb} \right ) \linebreak \left ( \frac{\BRd - \left (\ARdCb \right )
\left ( \frac{\BRb}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\BRc - \left (\ARcCb
\right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right )
\left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb
\right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right ) - \left (\frac{\ARbCc}{\ARbCb}
\right ) \linebreak \left ( \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left (
\frac{\ARbCc}{\ARbCb}\right )} - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right )
\left ( \frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left ( \frac{\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right ) - \left (\ARdCc
- \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{\ARcCc
- \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right ) - \left (\ARdCc
- \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc
- \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )\right )\\ \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )}{
\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} - \left (\frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )}{\ARcCc
- \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )} \right ) \linebreak \left ( \frac{\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}\right )
- \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}\right )
- \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{\ARbCb}
\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\right )\\ \frac{\BRd - \left (\ARdCb \right ) \left ( \frac{\BRb}{\ARbCb}
\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\BRc - \left (\ARcCb \right ) \left ( \frac{\BRb}{
\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}{\ARdCd - \left (\ARdCb \right ) \left ( \frac{\ARbCd}{\ARbCb}
\right ) - \left (\ARdCc - \left (\ARdCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right ) \right ) \left ( \frac{\ARcCd - \left (\ARcCb \right ) \left ( \frac{\ARbCd}{
\ARbCb}\right )}{\ARcCc - \left (\ARcCb \right ) \left ( \frac{\ARbCc}{\ARbCb}\right )}\right )}\end{pmatrix}\end{equation*}
\end{document}



Here is a link to the pdf showing all of the work for the inverse of a
3x3 matrix: