Sunday, July 20, 2014

Inverse of a 1x1 Matrix


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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 1x1 matrix is defined as follows. For a 1x1 matrix:

$$ A = \begin{pmatrix}a_{1,1}\end{pmatrix} $$

$$ A^{-1} = \begin{pmatrix}\frac{1}{a_{1,1}} \end {pmatrix} $$

Obviously, if $$a_{1,1} = 0$$ the matrix has no inverse.

The latex code generated for a 1x1 inverse is the following:

 
\documentclass{article}

% This is the output from LatexMatrixInverse 1.0 for a matrix with rank 1 
% For more information on this application, see
% 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={8.5in,11in}}

%Import AMS Latex packages
\usepackage{amsmath, amssymb}
\setcounter{MaxMatrixCols}{3}


%Variable definition

\begin{document}
% Definition of initial A
% A row 1
\newcommand{\ARbCb}{a_{1,1}}

% Definition of initial B
\newcommand{\BRb}{b_{1}}



LatexMatrixInverse 1.0 Output for rank 1, ShowIntermediateSteps is True.

For more information on this application, please see http://mikemstech.blogspot.com

Given the following initial matrices:
\begin{equation*}
A = \begin{pmatrix}\ARbCb
\end{pmatrix}B = \begin{pmatrix}\BRb\end{pmatrix}\end{equation*}
 We want to find $A^{-1}$ and $A^{-1} B$...
\begin{equation*}
\left ( \begin{array}{c|c|c}\ARbCb
&1
&\BRb\end{array} \right )
\end{equation*}
\begin{equation*}
\xrightarrow{\frac{1}{\ARbCb} R_{1}}
\left ( \begin{array}{c|c|c}1
&\frac{1}{\ARbCb}
&\frac{\BRb}{\ARbCb}\end{array} \right )
\end{equation*}
\begin{equation*}
A^{-1} = \begin{pmatrix}\frac{1}{\ARbCb}
\end{pmatrix}A^{-1}B = \begin{pmatrix}\frac{\BRb}{\ARbCb}\end{pmatrix}\end{equation*}
\end{document}


Here is a screenshot of the generated file that shows all of the steps for a 1x1 matrix inverse:





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