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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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