algebra.tex

\documentclass[10pt,a4paper,dvipdfmx]{article}
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\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{multicol}
\usepackage[hidelinks,dvipdfm]{hyperref}
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\title{M\'etodos para resoluci\'on de sistemas de ecuaciones lineales $LU$}
\author{Ricardo-Francisco Luis Mart\'inez}
\begin{document}
\maketitle
\tableofcontents
\newpage
\section{Metodo Lu}
\subsection{Caso de matrices de $2\times 2$ }
$$ \left( 
\begin{array}{cc}
a_{{1}{1}} & a_{{1}{2}} \\
a_{{2}{1}} & a_{{2}{2}} 
 \end{array}
\right)
 = \left( 
\begin{array}{cc}
l_{{1}{1}} & 0 \\
l_{{2}{1}} & l_{{2}{2}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{cc}
1 & u_{{1}{2}} \\
0 & 1 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{cc}
l_{{1}{1}} & u_{{1}{2}} \\
l_{{2}{1}} & l_{{2}{2}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{1}{1}} = l_{{1}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} $$
$$ a_{{1}{2}} = l_{{1}{1}} u_{{1}{2}} $$
$$ a_{{2}{2}} = l_{{2}{1}} u_{{1}{2}} + l_{{2}{2}} $$
\vfill\null
\columnbreak
$$ l_{{1}{1}} = a_{{1}{1}} $$
$$ l_{{2}{1}} = a_{{2}{1}} $$
$$ u_{{1}{2}} = \dfrac{a_{{1}{2}}}{l_{{1}{1}}} $$
$$ l_{{2}{2}} = a_{{2}{2}}- l_{{2}{1}} u_{{1}{2}} $$
\end{multicols}
\subsection{Caso de matrices de $3\times 3$ }
$$ \left( 
\begin{array}{ccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccc}
l_{{1}{1}} & 0 & 0 \\
l_{{2}{1}} & l_{{2}{2}} & 0 \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{ccc}
1 & u_{{1}{2}} & u_{{1}{3}} \\
0 & 1 & u_{{2}{3}} \\
0 & 0 & 1 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{ccc}
l_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} \\
l_{{2}{1}} & l_{{2}{2}} & u_{{2}{3}} \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{1}{1}} = l_{{1}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} $$
$$ a_{{3}{1}} = l_{{3}{1}} $$
$$ a_{{1}{2}} = l_{{1}{1}} u_{{1}{2}} $$
$$ a_{{2}{2}} = l_{{2}{1}} u_{{1}{2}} + l_{{2}{2}} $$
$$ a_{{3}{2}} = l_{{3}{1}} u_{{1}{2}} + l_{{3}{2}} $$
$$ a_{{1}{3}} = l_{{1}{1}} u_{{1}{3}} $$
$$ a_{{2}{3}} = l_{{2}{1}} u_{{1}{3}} + l_{{2}{2}} u_{{2}{3}} $$
$$ a_{{3}{3}} = l_{{3}{1}} u_{{1}{3}} + l_{{3}{2}} u_{{2}{3}} + l_{{3}{3}} $$
\vfill\null
\columnbreak
$$ l_{{1}{1}} = a_{{1}{1}} $$
$$ l_{{2}{1}} = a_{{2}{1}} $$
$$ l_{{3}{1}} = a_{{3}{1}} $$
$$ u_{{1}{2}} = \dfrac{a_{{1}{2}}}{l_{{1}{1}}} $$
$$ l_{{2}{2}} = a_{{2}{2}}- l_{{2}{1}} u_{{1}{2}} $$
$$ l_{{3}{2}} = a_{{3}{2}}- l_{{3}{1}} u_{{1}{2}} $$
$$ u_{{1}{3}} = \dfrac{a_{{1}{3}}}{l_{{1}{1}}} $$
$$ u_{{2}{3}} = \dfrac{a_{{2}{3}}- l_{{2}{1}} u_{{1}{3}}}{l_{{2}{2}}} $$
$$ l_{{3}{3}} = a_{{3}{3}}- l_{{3}{1}} u_{{1}{3}}- l_{{3}{2}} u_{{2}{3}} $$
\end{multicols}
\subsection{Caso de matrices de $4\times 4$ }
$$ \left( 
\begin{array}{cccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} & a_{{1}{4}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} & a_{{2}{4}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} & a_{{3}{4}} \\
a_{{4}{1}} & a_{{4}{2}} & a_{{4}{3}} & a_{{4}{4}} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccc}
l_{{1}{1}} & 0 & 0 & 0 \\
l_{{2}{1}} & l_{{2}{2}} & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{cccc}
1 & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} \\
0 & 1 & u_{{2}{3}} & u_{{2}{4}} \\
0 & 0 & 1 & u_{{3}{4}} \\
0 & 0 & 0 & 1 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{cccc}
l_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} \\
l_{{2}{1}} & l_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & u_{{3}{4}} \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{1}{1}} = l_{{1}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} $$
$$ a_{{3}{1}} = l_{{3}{1}} $$
$$ a_{{4}{1}} = l_{{4}{1}} $$
$$ a_{{1}{2}} = l_{{1}{1}} u_{{1}{2}} $$
$$ a_{{2}{2}} = l_{{2}{1}} u_{{1}{2}} + l_{{2}{2}} $$
$$ a_{{3}{2}} = l_{{3}{1}} u_{{1}{2}} + l_{{3}{2}} $$
$$ a_{{4}{2}} = l_{{4}{1}} u_{{1}{2}} + l_{{4}{2}} $$
$$ a_{{1}{3}} = l_{{1}{1}} u_{{1}{3}} $$
$$ a_{{2}{3}} = l_{{2}{1}} u_{{1}{3}} + l_{{2}{2}} u_{{2}{3}} $$
$$ a_{{3}{3}} = l_{{3}{1}} u_{{1}{3}} + l_{{3}{2}} u_{{2}{3}} + l_{{3}{3}} $$
$$ a_{{4}{3}} = l_{{4}{1}} u_{{1}{3}} + l_{{4}{2}} u_{{2}{3}} + l_{{4}{3}} $$
$$ a_{{1}{4}} = l_{{1}{1}} u_{{1}{4}} $$
$$ a_{{2}{4}} = l_{{2}{1}} u_{{1}{4}} + l_{{2}{2}} u_{{2}{4}} $$
$$ a_{{3}{4}} = l_{{3}{1}} u_{{1}{4}} + l_{{3}{2}} u_{{2}{4}} + l_{{3}{3}} u_{{3}{4}} $$
$$ a_{{4}{4}} = l_{{4}{1}} u_{{1}{4}} + l_{{4}{2}} u_{{2}{4}} + l_{{4}{3}} u_{{3}{4}} + l_{{4}{4}} $$
\vfill\null
\columnbreak
$$ l_{{1}{1}} = a_{{1}{1}} $$
$$ l_{{2}{1}} = a_{{2}{1}} $$
$$ l_{{3}{1}} = a_{{3}{1}} $$
$$ l_{{4}{1}} = a_{{4}{1}} $$
$$ u_{{1}{2}} = \dfrac{a_{{1}{2}}}{l_{{1}{1}}} $$
$$ l_{{2}{2}} = a_{{2}{2}}- l_{{2}{1}} u_{{1}{2}} $$
$$ l_{{3}{2}} = a_{{3}{2}}- l_{{3}{1}} u_{{1}{2}} $$
$$ l_{{4}{2}} = a_{{4}{2}}- l_{{4}{1}} u_{{1}{2}} $$
$$ u_{{1}{3}} = \dfrac{a_{{1}{3}}}{l_{{1}{1}}} $$
$$ u_{{2}{3}} = \dfrac{a_{{2}{3}}- l_{{2}{1}} u_{{1}{3}}}{l_{{2}{2}}} $$
$$ l_{{3}{3}} = a_{{3}{3}}- l_{{3}{1}} u_{{1}{3}}- l_{{3}{2}} u_{{2}{3}} $$
$$ l_{{4}{3}} = a_{{4}{3}}- l_{{4}{1}} u_{{1}{3}}- l_{{4}{2}} u_{{2}{3}} $$
$$ u_{{1}{4}} = \dfrac{a_{{1}{4}}}{l_{{1}{1}}} $$
$$ u_{{2}{4}} = \dfrac{a_{{2}{4}}- l_{{2}{1}} u_{{1}{4}}}{l_{{2}{2}}} $$
$$ u_{{3}{4}} = \dfrac{a_{{3}{4}}- l_{{3}{1}} u_{{1}{4}}- l_{{3}{2}} u_{{2}{4}}}{l_{{3}{3}}} $$
$$ l_{{4}{4}} = a_{{4}{4}}- l_{{4}{1}} u_{{1}{4}}- l_{{4}{2}} u_{{2}{4}}- l_{{4}{3}} u_{{3}{4}} $$
\end{multicols}
\subsection{Caso de matrices de $5\times 5$ }
$$ \left( 
\begin{array}{ccccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} & a_{{1}{4}} & a_{{1}{5}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} & a_{{2}{4}} & a_{{2}{5}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} & a_{{3}{4}} & a_{{3}{5}} \\
a_{{4}{1}} & a_{{4}{2}} & a_{{4}{3}} & a_{{4}{4}} & a_{{4}{5}} \\
a_{{5}{1}} & a_{{5}{2}} & a_{{5}{3}} & a_{{5}{4}} & a_{{5}{5}} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccccc}
l_{{1}{1}} & 0 & 0 & 0 & 0 \\
l_{{2}{1}} & l_{{2}{2}} & 0 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & 0 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} & 0 \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & l_{{5}{5}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{ccccc}
1 & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} \\
0 & 1 & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} \\
0 & 0 & 1 & u_{{3}{4}} & u_{{3}{5}} \\
0 & 0 & 0 & 1 & u_{{4}{5}} \\
0 & 0 & 0 & 0 & 1 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{ccccc}
l_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} \\
l_{{2}{1}} & l_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} & u_{{4}{5}} \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & l_{{5}{5}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{1}{1}} = l_{{1}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} $$
$$ a_{{3}{1}} = l_{{3}{1}} $$
$$ a_{{4}{1}} = l_{{4}{1}} $$
$$ a_{{5}{1}} = l_{{5}{1}} $$
$$ a_{{1}{2}} = l_{{1}{1}} u_{{1}{2}} $$
$$ a_{{2}{2}} = l_{{2}{1}} u_{{1}{2}} + l_{{2}{2}} $$
$$ a_{{3}{2}} = l_{{3}{1}} u_{{1}{2}} + l_{{3}{2}} $$
$$ a_{{4}{2}} = l_{{4}{1}} u_{{1}{2}} + l_{{4}{2}} $$
$$ a_{{5}{2}} = l_{{5}{1}} u_{{1}{2}} + l_{{5}{2}} $$
$$ a_{{1}{3}} = l_{{1}{1}} u_{{1}{3}} $$
$$ a_{{2}{3}} = l_{{2}{1}} u_{{1}{3}} + l_{{2}{2}} u_{{2}{3}} $$
$$ a_{{3}{3}} = l_{{3}{1}} u_{{1}{3}} + l_{{3}{2}} u_{{2}{3}} + l_{{3}{3}} $$
$$ a_{{4}{3}} = l_{{4}{1}} u_{{1}{3}} + l_{{4}{2}} u_{{2}{3}} + l_{{4}{3}} $$
$$ a_{{5}{3}} = l_{{5}{1}} u_{{1}{3}} + l_{{5}{2}} u_{{2}{3}} + l_{{5}{3}} $$
$$ a_{{1}{4}} = l_{{1}{1}} u_{{1}{4}} $$
$$ a_{{2}{4}} = l_{{2}{1}} u_{{1}{4}} + l_{{2}{2}} u_{{2}{4}} $$
$$ a_{{3}{4}} = l_{{3}{1}} u_{{1}{4}} + l_{{3}{2}} u_{{2}{4}} + l_{{3}{3}} u_{{3}{4}} $$
$$ a_{{4}{4}} = l_{{4}{1}} u_{{1}{4}} + l_{{4}{2}} u_{{2}{4}} + l_{{4}{3}} u_{{3}{4}} + l_{{4}{4}} $$
$$ a_{{5}{4}} = l_{{5}{1}} u_{{1}{4}} + l_{{5}{2}} u_{{2}{4}} + l_{{5}{3}} u_{{3}{4}} + l_{{5}{4}} $$
$$ a_{{1}{5}} = l_{{1}{1}} u_{{1}{5}} $$
$$ a_{{2}{5}} = l_{{2}{1}} u_{{1}{5}} + l_{{2}{2}} u_{{2}{5}} $$
$$ a_{{3}{5}} = l_{{3}{1}} u_{{1}{5}} + l_{{3}{2}} u_{{2}{5}} + l_{{3}{3}} u_{{3}{5}} $$
$$ a_{{4}{5}} = l_{{4}{1}} u_{{1}{5}} + l_{{4}{2}} u_{{2}{5}} + l_{{4}{3}} u_{{3}{5}} + l_{{4}{4}} u_{{4}{5}} $$
$$ a_{{5}{5}} = l_{{5}{1}} u_{{1}{5}} + l_{{5}{2}} u_{{2}{5}} + l_{{5}{3}} u_{{3}{5}} + l_{{5}{4}} u_{{4}{5}} + l_{{5}{5}} $$
\vfill\null
\columnbreak
$$ l_{{1}{1}} = a_{{1}{1}} $$
$$ l_{{2}{1}} = a_{{2}{1}} $$
$$ l_{{3}{1}} = a_{{3}{1}} $$
$$ l_{{4}{1}} = a_{{4}{1}} $$
$$ l_{{5}{1}} = a_{{5}{1}} $$
$$ u_{{1}{2}} = \dfrac{a_{{1}{2}}}{l_{{1}{1}}} $$
$$ l_{{2}{2}} = a_{{2}{2}}- l_{{2}{1}} u_{{1}{2}} $$
$$ l_{{3}{2}} = a_{{3}{2}}- l_{{3}{1}} u_{{1}{2}} $$
$$ l_{{4}{2}} = a_{{4}{2}}- l_{{4}{1}} u_{{1}{2}} $$
$$ l_{{5}{2}} = a_{{5}{2}}- l_{{5}{1}} u_{{1}{2}} $$
$$ u_{{1}{3}} = \dfrac{a_{{1}{3}}}{l_{{1}{1}}} $$
$$ u_{{2}{3}} = \dfrac{a_{{2}{3}}- l_{{2}{1}} u_{{1}{3}}}{l_{{2}{2}}} $$
$$ l_{{3}{3}} = a_{{3}{3}}- l_{{3}{1}} u_{{1}{3}}- l_{{3}{2}} u_{{2}{3}} $$
$$ l_{{4}{3}} = a_{{4}{3}}- l_{{4}{1}} u_{{1}{3}}- l_{{4}{2}} u_{{2}{3}} $$
$$ l_{{5}{3}} = a_{{5}{3}}- l_{{5}{1}} u_{{1}{3}}- l_{{5}{2}} u_{{2}{3}} $$
$$ u_{{1}{4}} = \dfrac{a_{{1}{4}}}{l_{{1}{1}}} $$
$$ u_{{2}{4}} = \dfrac{a_{{2}{4}}- l_{{2}{1}} u_{{1}{4}}}{l_{{2}{2}}} $$
$$ u_{{3}{4}} = \dfrac{a_{{3}{4}}- l_{{3}{1}} u_{{1}{4}}- l_{{3}{2}} u_{{2}{4}}}{l_{{3}{3}}} $$
$$ l_{{4}{4}} = a_{{4}{4}}- l_{{4}{1}} u_{{1}{4}}- l_{{4}{2}} u_{{2}{4}}- l_{{4}{3}} u_{{3}{4}} $$
$$ l_{{5}{4}} = a_{{5}{4}}- l_{{5}{1}} u_{{1}{4}}- l_{{5}{2}} u_{{2}{4}}- l_{{5}{3}} u_{{3}{4}} $$
$$ u_{{1}{5}} = \dfrac{a_{{1}{5}}}{l_{{1}{1}}} $$
$$ u_{{2}{5}} = \dfrac{a_{{2}{5}}- l_{{2}{1}} u_{{1}{5}}}{l_{{2}{2}}} $$
$$ u_{{3}{5}} = \dfrac{a_{{3}{5}}- l_{{3}{1}} u_{{1}{5}}- l_{{3}{2}} u_{{2}{5}}}{l_{{3}{3}}} $$
$$ u_{{4}{5}} = \dfrac{a_{{4}{5}}- l_{{4}{1}} u_{{1}{5}}- l_{{4}{2}} u_{{2}{5}}- l_{{4}{3}} u_{{3}{5}}}{l_{{4}{4}}} $$
$$ l_{{5}{5}} = a_{{5}{5}}- l_{{5}{1}} u_{{1}{5}}- l_{{5}{2}} u_{{2}{5}}- l_{{5}{3}} u_{{3}{5}}- l_{{5}{4}} u_{{4}{5}} $$
\end{multicols}
\subsection{Caso de matrices de $6\times 6$ }
$$ \left( 
\begin{array}{cccccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} & a_{{1}{4}} & a_{{1}{5}} & a_{{1}{6}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} & a_{{2}{4}} & a_{{2}{5}} & a_{{2}{6}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} & a_{{3}{4}} & a_{{3}{5}} & a_{{3}{6}} \\
a_{{4}{1}} & a_{{4}{2}} & a_{{4}{3}} & a_{{4}{4}} & a_{{4}{5}} & a_{{4}{6}} \\
a_{{5}{1}} & a_{{5}{2}} & a_{{5}{3}} & a_{{5}{4}} & a_{{5}{5}} & a_{{5}{6}} \\
a_{{6}{1}} & a_{{6}{2}} & a_{{6}{3}} & a_{{6}{4}} & a_{{6}{5}} & a_{{6}{6}} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccccc}
l_{{1}{1}} & 0 & 0 & 0 & 0 & 0 \\
l_{{2}{1}} & l_{{2}{2}} & 0 & 0 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & 0 & 0 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} & 0 & 0 \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & l_{{5}{5}} & 0 \\
l_{{6}{1}} & l_{{6}{2}} & l_{{6}{3}} & l_{{6}{4}} & l_{{6}{5}} & l_{{6}{6}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{cccccc}
1 & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} & u_{{1}{6}} \\
0 & 1 & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} & u_{{2}{6}} \\
0 & 0 & 1 & u_{{3}{4}} & u_{{3}{5}} & u_{{3}{6}} \\
0 & 0 & 0 & 1 & u_{{4}{5}} & u_{{4}{6}} \\
0 & 0 & 0 & 0 & 1 & u_{{5}{6}} \\
0 & 0 & 0 & 0 & 0 & 1 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{cccccc}
l_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} & u_{{1}{6}} \\
l_{{2}{1}} & l_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} & u_{{2}{6}} \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} & u_{{3}{6}} \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} & u_{{4}{5}} & u_{{4}{6}} \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & l_{{5}{5}} & u_{{5}{6}} \\
l_{{6}{1}} & l_{{6}{2}} & l_{{6}{3}} & l_{{6}{4}} & l_{{6}{5}} & l_{{6}{6}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{1}{1}} = l_{{1}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} $$
$$ a_{{3}{1}} = l_{{3}{1}} $$
$$ a_{{4}{1}} = l_{{4}{1}} $$
$$ a_{{5}{1}} = l_{{5}{1}} $$
$$ a_{{6}{1}} = l_{{6}{1}} $$
$$ a_{{1}{2}} = l_{{1}{1}} u_{{1}{2}} $$
$$ a_{{2}{2}} = l_{{2}{1}} u_{{1}{2}} + l_{{2}{2}} $$
$$ a_{{3}{2}} = l_{{3}{1}} u_{{1}{2}} + l_{{3}{2}} $$
$$ a_{{4}{2}} = l_{{4}{1}} u_{{1}{2}} + l_{{4}{2}} $$
$$ a_{{5}{2}} = l_{{5}{1}} u_{{1}{2}} + l_{{5}{2}} $$
$$ a_{{6}{2}} = l_{{6}{1}} u_{{1}{2}} + l_{{6}{2}} $$
$$ a_{{1}{3}} = l_{{1}{1}} u_{{1}{3}} $$
$$ a_{{2}{3}} = l_{{2}{1}} u_{{1}{3}} + l_{{2}{2}} u_{{2}{3}} $$
$$ a_{{3}{3}} = l_{{3}{1}} u_{{1}{3}} + l_{{3}{2}} u_{{2}{3}} + l_{{3}{3}} $$
$$ a_{{4}{3}} = l_{{4}{1}} u_{{1}{3}} + l_{{4}{2}} u_{{2}{3}} + l_{{4}{3}} $$
$$ a_{{5}{3}} = l_{{5}{1}} u_{{1}{3}} + l_{{5}{2}} u_{{2}{3}} + l_{{5}{3}} $$
$$ a_{{6}{3}} = l_{{6}{1}} u_{{1}{3}} + l_{{6}{2}} u_{{2}{3}} + l_{{6}{3}} $$
$$ a_{{1}{4}} = l_{{1}{1}} u_{{1}{4}} $$
$$ a_{{2}{4}} = l_{{2}{1}} u_{{1}{4}} + l_{{2}{2}} u_{{2}{4}} $$
$$ a_{{3}{4}} = l_{{3}{1}} u_{{1}{4}} + l_{{3}{2}} u_{{2}{4}} + l_{{3}{3}} u_{{3}{4}} $$
$$ a_{{4}{4}} = l_{{4}{1}} u_{{1}{4}} + l_{{4}{2}} u_{{2}{4}} + l_{{4}{3}} u_{{3}{4}} + l_{{4}{4}} $$
$$ a_{{5}{4}} = l_{{5}{1}} u_{{1}{4}} + l_{{5}{2}} u_{{2}{4}} + l_{{5}{3}} u_{{3}{4}} + l_{{5}{4}} $$
$$ a_{{6}{4}} = l_{{6}{1}} u_{{1}{4}} + l_{{6}{2}} u_{{2}{4}} + l_{{6}{3}} u_{{3}{4}} + l_{{6}{4}} $$
$$ a_{{1}{5}} = l_{{1}{1}} u_{{1}{5}} $$
$$ a_{{2}{5}} = l_{{2}{1}} u_{{1}{5}} + l_{{2}{2}} u_{{2}{5}} $$
$$ a_{{3}{5}} = l_{{3}{1}} u_{{1}{5}} + l_{{3}{2}} u_{{2}{5}} + l_{{3}{3}} u_{{3}{5}} $$
$$ a_{{4}{5}} = l_{{4}{1}} u_{{1}{5}} + l_{{4}{2}} u_{{2}{5}} + l_{{4}{3}} u_{{3}{5}} + l_{{4}{4}} u_{{4}{5}} $$
$$ a_{{5}{5}} = l_{{5}{1}} u_{{1}{5}} + l_{{5}{2}} u_{{2}{5}} + l_{{5}{3}} u_{{3}{5}} + l_{{5}{4}} u_{{4}{5}} + l_{{5}{5}} $$
$$ a_{{6}{5}} = l_{{6}{1}} u_{{1}{5}} + l_{{6}{2}} u_{{2}{5}} + l_{{6}{3}} u_{{3}{5}} + l_{{6}{4}} u_{{4}{5}} + l_{{6}{5}} $$
$$ a_{{1}{6}} = l_{{1}{1}} u_{{1}{6}} $$
$$ a_{{2}{6}} = l_{{2}{1}} u_{{1}{6}} + l_{{2}{2}} u_{{2}{6}} $$
$$ a_{{3}{6}} = l_{{3}{1}} u_{{1}{6}} + l_{{3}{2}} u_{{2}{6}} + l_{{3}{3}} u_{{3}{6}} $$
$$ a_{{4}{6}} = l_{{4}{1}} u_{{1}{6}} + l_{{4}{2}} u_{{2}{6}} + l_{{4}{3}} u_{{3}{6}} + l_{{4}{4}} u_{{4}{6}} $$
$$ a_{{5}{6}} = l_{{5}{1}} u_{{1}{6}} + l_{{5}{2}} u_{{2}{6}} + l_{{5}{3}} u_{{3}{6}} + l_{{5}{4}} u_{{4}{6}} + l_{{5}{5}} u_{{5}{6}} $$
$$ a_{{6}{6}} = l_{{6}{1}} u_{{1}{6}} + l_{{6}{2}} u_{{2}{6}} + l_{{6}{3}} u_{{3}{6}} + l_{{6}{4}} u_{{4}{6}} + l_{{6}{5}} u_{{5}{6}} + l_{{6}{6}} $$
\vfill\null
\columnbreak
$$ l_{{1}{1}} = a_{{1}{1}} $$
$$ l_{{2}{1}} = a_{{2}{1}} $$
$$ l_{{3}{1}} = a_{{3}{1}} $$
$$ l_{{4}{1}} = a_{{4}{1}} $$
$$ l_{{5}{1}} = a_{{5}{1}} $$
$$ l_{{6}{1}} = a_{{6}{1}} $$
$$ u_{{1}{2}} = \dfrac{a_{{1}{2}}}{l_{{1}{1}}} $$
$$ l_{{2}{2}} = a_{{2}{2}}- l_{{2}{1}} u_{{1}{2}} $$
$$ l_{{3}{2}} = a_{{3}{2}}- l_{{3}{1}} u_{{1}{2}} $$
$$ l_{{4}{2}} = a_{{4}{2}}- l_{{4}{1}} u_{{1}{2}} $$
$$ l_{{5}{2}} = a_{{5}{2}}- l_{{5}{1}} u_{{1}{2}} $$
$$ l_{{6}{2}} = a_{{6}{2}}- l_{{6}{1}} u_{{1}{2}} $$
$$ u_{{1}{3}} = \dfrac{a_{{1}{3}}}{l_{{1}{1}}} $$
$$ u_{{2}{3}} = \dfrac{a_{{2}{3}}- l_{{2}{1}} u_{{1}{3}}}{l_{{2}{2}}} $$
$$ l_{{3}{3}} = a_{{3}{3}}- l_{{3}{1}} u_{{1}{3}}- l_{{3}{2}} u_{{2}{3}} $$
$$ l_{{4}{3}} = a_{{4}{3}}- l_{{4}{1}} u_{{1}{3}}- l_{{4}{2}} u_{{2}{3}} $$
$$ l_{{5}{3}} = a_{{5}{3}}- l_{{5}{1}} u_{{1}{3}}- l_{{5}{2}} u_{{2}{3}} $$
$$ l_{{6}{3}} = a_{{6}{3}}- l_{{6}{1}} u_{{1}{3}}- l_{{6}{2}} u_{{2}{3}} $$
$$ u_{{1}{4}} = \dfrac{a_{{1}{4}}}{l_{{1}{1}}} $$
$$ u_{{2}{4}} = \dfrac{a_{{2}{4}}- l_{{2}{1}} u_{{1}{4}}}{l_{{2}{2}}} $$
$$ u_{{3}{4}} = \dfrac{a_{{3}{4}}- l_{{3}{1}} u_{{1}{4}}- l_{{3}{2}} u_{{2}{4}}}{l_{{3}{3}}} $$
$$ l_{{4}{4}} = a_{{4}{4}}- l_{{4}{1}} u_{{1}{4}}- l_{{4}{2}} u_{{2}{4}}- l_{{4}{3}} u_{{3}{4}} $$
$$ l_{{5}{4}} = a_{{5}{4}}- l_{{5}{1}} u_{{1}{4}}- l_{{5}{2}} u_{{2}{4}}- l_{{5}{3}} u_{{3}{4}} $$
$$ l_{{6}{4}} = a_{{6}{4}}- l_{{6}{1}} u_{{1}{4}}- l_{{6}{2}} u_{{2}{4}}- l_{{6}{3}} u_{{3}{4}} $$
$$ u_{{1}{5}} = \dfrac{a_{{1}{5}}}{l_{{1}{1}}} $$
$$ u_{{2}{5}} = \dfrac{a_{{2}{5}}- l_{{2}{1}} u_{{1}{5}}}{l_{{2}{2}}} $$
$$ u_{{3}{5}} = \dfrac{a_{{3}{5}}- l_{{3}{1}} u_{{1}{5}}- l_{{3}{2}} u_{{2}{5}}}{l_{{3}{3}}} $$
$$ u_{{4}{5}} = \dfrac{a_{{4}{5}}- l_{{4}{1}} u_{{1}{5}}- l_{{4}{2}} u_{{2}{5}}- l_{{4}{3}} u_{{3}{5}}}{l_{{4}{4}}} $$
$$ l_{{5}{5}} = a_{{5}{5}}- l_{{5}{1}} u_{{1}{5}}- l_{{5}{2}} u_{{2}{5}}- l_{{5}{3}} u_{{3}{5}}- l_{{5}{4}} u_{{4}{5}} $$
$$ l_{{6}{5}} = a_{{6}{5}}- l_{{6}{1}} u_{{1}{5}}- l_{{6}{2}} u_{{2}{5}}- l_{{6}{3}} u_{{3}{5}}- l_{{6}{4}} u_{{4}{5}} $$
$$ u_{{1}{6}} = \dfrac{a_{{1}{6}}}{l_{{1}{1}}} $$
$$ u_{{2}{6}} = \dfrac{a_{{2}{6}}- l_{{2}{1}} u_{{1}{6}}}{l_{{2}{2}}} $$
$$ u_{{3}{6}} = \dfrac{a_{{3}{6}}- l_{{3}{1}} u_{{1}{6}}- l_{{3}{2}} u_{{2}{6}}}{l_{{3}{3}}} $$
$$ u_{{4}{6}} = \dfrac{a_{{4}{6}}- l_{{4}{1}} u_{{1}{6}}- l_{{4}{2}} u_{{2}{6}}- l_{{4}{3}} u_{{3}{6}}}{l_{{4}{4}}} $$
$$ u_{{5}{6}} = \dfrac{a_{{5}{6}}- l_{{5}{1}} u_{{1}{6}}- l_{{5}{2}} u_{{2}{6}}- l_{{5}{3}} u_{{3}{6}}- l_{{5}{4}} u_{{4}{6}}}{l_{{5}{5}}} $$
$$ l_{{6}{6}} = a_{{6}{6}}- l_{{6}{1}} u_{{1}{6}}- l_{{6}{2}} u_{{2}{6}}- l_{{6}{3}} u_{{3}{6}}- l_{{6}{4}} u_{{4}{6}}- l_{{6}{5}} u_{{5}{6}} $$
\end{multicols}
\section{Metodo lU}
\subsection{Caso de matrices de $2\times 2$ }
$$ \left( 
\begin{array}{cc}
a_{{1}{1}} & a_{{1}{2}} \\
a_{{2}{1}} & a_{{2}{2}} 
 \end{array}
\right)
 = \left( 
\begin{array}{cc}
1 & 0 \\
l_{{2}{1}} & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{cc}
u_{{1}{1}} & u_{{1}{2}} \\
0 & u_{{2}{2}} 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{cc}
u_{{1}{1}} & u_{{1}{2}} \\
l_{{2}{1}} & u_{{2}{2}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{1}{1}} = u_{{1}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} u_{{1}{1}} $$
$$ a_{{1}{2}} = u_{{1}{2}} $$
$$ a_{{2}{2}} = l_{{2}{1}} u_{{1}{2}} + u_{{2}{2}} $$
\vfill\null
\columnbreak
$$ u_{{1}{1}} = a_{{1}{1}} $$
$$ l_{{2}{1}} = \dfrac{a_{{2}{1}}}{u_{{1}{1}}} $$
$$ u_{{1}{2}} = a_{{1}{2}} $$
$$ u_{{2}{2}} = a_{{2}{2}}- l_{{2}{1}} u_{{1}{2}} $$
\end{multicols}
\subsection{Caso de matrices de $3\times 3$ }
$$ \left( 
\begin{array}{ccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccc}
1 & 0 & 0 \\
l_{{2}{1}} & 1 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{ccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} \\
0 & u_{{2}{2}} & u_{{2}{3}} \\
0 & 0 & u_{{3}{3}} 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{ccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} \\
l_{{2}{1}} & u_{{2}{2}} & u_{{2}{3}} \\
l_{{3}{1}} & l_{{3}{2}} & u_{{3}{3}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{1}{1}} = u_{{1}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} u_{{1}{1}} $$
$$ a_{{3}{1}} = l_{{3}{1}} u_{{1}{1}} $$
$$ a_{{1}{2}} = u_{{1}{2}} $$
$$ a_{{2}{2}} = l_{{2}{1}} u_{{1}{2}} + u_{{2}{2}} $$
$$ a_{{3}{2}} = l_{{3}{1}} u_{{1}{2}} + l_{{3}{2}} u_{{2}{2}} $$
$$ a_{{1}{3}} = u_{{1}{3}} $$
$$ a_{{2}{3}} = l_{{2}{1}} u_{{1}{3}} + u_{{2}{3}} $$
$$ a_{{3}{3}} = l_{{3}{1}} u_{{1}{3}} + l_{{3}{2}} u_{{2}{3}} + u_{{3}{3}} $$
\vfill\null
\columnbreak
$$ u_{{1}{1}} = a_{{1}{1}} $$
$$ l_{{2}{1}} = \dfrac{a_{{2}{1}}}{u_{{1}{1}}} $$
$$ l_{{3}{1}} = \dfrac{a_{{3}{1}}}{u_{{1}{1}}} $$
$$ u_{{1}{2}} = a_{{1}{2}} $$
$$ u_{{2}{2}} = a_{{2}{2}}- l_{{2}{1}} u_{{1}{2}} $$
$$ l_{{3}{2}} = \dfrac{a_{{3}{2}}- l_{{3}{1}} u_{{1}{2}}}{u_{{2}{2}}} $$
$$ u_{{1}{3}} = a_{{1}{3}} $$
$$ u_{{2}{3}} = a_{{2}{3}}- l_{{2}{1}} u_{{1}{3}} $$
$$ u_{{3}{3}} = a_{{3}{3}}- l_{{3}{1}} u_{{1}{3}}- l_{{3}{2}} u_{{2}{3}} $$
\end{multicols}
\subsection{Caso de matrices de $4\times 4$ }
$$ \left( 
\begin{array}{cccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} & a_{{1}{4}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} & a_{{2}{4}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} & a_{{3}{4}} \\
a_{{4}{1}} & a_{{4}{2}} & a_{{4}{3}} & a_{{4}{4}} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
l_{{2}{1}} & 1 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & 1 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{cccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} \\
0 & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} \\
0 & 0 & u_{{3}{3}} & u_{{3}{4}} \\
0 & 0 & 0 & u_{{4}{4}} 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{cccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} \\
l_{{2}{1}} & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} \\
l_{{3}{1}} & l_{{3}{2}} & u_{{3}{3}} & u_{{3}{4}} \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & u_{{4}{4}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{1}{1}} = u_{{1}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} u_{{1}{1}} $$
$$ a_{{3}{1}} = l_{{3}{1}} u_{{1}{1}} $$
$$ a_{{4}{1}} = l_{{4}{1}} u_{{1}{1}} $$
$$ a_{{1}{2}} = u_{{1}{2}} $$
$$ a_{{2}{2}} = l_{{2}{1}} u_{{1}{2}} + u_{{2}{2}} $$
$$ a_{{3}{2}} = l_{{3}{1}} u_{{1}{2}} + l_{{3}{2}} u_{{2}{2}} $$
$$ a_{{4}{2}} = l_{{4}{1}} u_{{1}{2}} + l_{{4}{2}} u_{{2}{2}} $$
$$ a_{{1}{3}} = u_{{1}{3}} $$
$$ a_{{2}{3}} = l_{{2}{1}} u_{{1}{3}} + u_{{2}{3}} $$
$$ a_{{3}{3}} = l_{{3}{1}} u_{{1}{3}} + l_{{3}{2}} u_{{2}{3}} + u_{{3}{3}} $$
$$ a_{{4}{3}} = l_{{4}{1}} u_{{1}{3}} + l_{{4}{2}} u_{{2}{3}} + l_{{4}{3}} u_{{3}{3}} $$
$$ a_{{1}{4}} = u_{{1}{4}} $$
$$ a_{{2}{4}} = l_{{2}{1}} u_{{1}{4}} + u_{{2}{4}} $$
$$ a_{{3}{4}} = l_{{3}{1}} u_{{1}{4}} + l_{{3}{2}} u_{{2}{4}} + u_{{3}{4}} $$
$$ a_{{4}{4}} = l_{{4}{1}} u_{{1}{4}} + l_{{4}{2}} u_{{2}{4}} + l_{{4}{3}} u_{{3}{4}} + u_{{4}{4}} $$
\vfill\null
\columnbreak
$$ u_{{1}{1}} = a_{{1}{1}} $$
$$ l_{{2}{1}} = \dfrac{a_{{2}{1}}}{u_{{1}{1}}} $$
$$ l_{{3}{1}} = \dfrac{a_{{3}{1}}}{u_{{1}{1}}} $$
$$ l_{{4}{1}} = \dfrac{a_{{4}{1}}}{u_{{1}{1}}} $$
$$ u_{{1}{2}} = a_{{1}{2}} $$
$$ u_{{2}{2}} = a_{{2}{2}}- l_{{2}{1}} u_{{1}{2}} $$
$$ l_{{3}{2}} = \dfrac{a_{{3}{2}}- l_{{3}{1}} u_{{1}{2}}}{u_{{2}{2}}} $$
$$ l_{{4}{2}} = \dfrac{a_{{4}{2}}- l_{{4}{1}} u_{{1}{2}}}{u_{{2}{2}}} $$
$$ u_{{1}{3}} = a_{{1}{3}} $$
$$ u_{{2}{3}} = a_{{2}{3}}- l_{{2}{1}} u_{{1}{3}} $$
$$ u_{{3}{3}} = a_{{3}{3}}- l_{{3}{1}} u_{{1}{3}}- l_{{3}{2}} u_{{2}{3}} $$
$$ l_{{4}{3}} = \dfrac{a_{{4}{3}}- l_{{4}{1}} u_{{1}{3}}- l_{{4}{2}} u_{{2}{3}}}{u_{{3}{3}}} $$
$$ u_{{1}{4}} = a_{{1}{4}} $$
$$ u_{{2}{4}} = a_{{2}{4}}- l_{{2}{1}} u_{{1}{4}} $$
$$ u_{{3}{4}} = a_{{3}{4}}- l_{{3}{1}} u_{{1}{4}}- l_{{3}{2}} u_{{2}{4}} $$
$$ u_{{4}{4}} = a_{{4}{4}}- l_{{4}{1}} u_{{1}{4}}- l_{{4}{2}} u_{{2}{4}}- l_{{4}{3}} u_{{3}{4}} $$
\end{multicols}
\subsection{Caso de matrices de $5\times 5$ }
$$ \left( 
\begin{array}{ccccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} & a_{{1}{4}} & a_{{1}{5}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} & a_{{2}{4}} & a_{{2}{5}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} & a_{{3}{4}} & a_{{3}{5}} \\
a_{{4}{1}} & a_{{4}{2}} & a_{{4}{3}} & a_{{4}{4}} & a_{{4}{5}} \\
a_{{5}{1}} & a_{{5}{2}} & a_{{5}{3}} & a_{{5}{4}} & a_{{5}{5}} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
l_{{2}{1}} & 1 & 0 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & 1 & 0 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & 1 & 0 \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{ccccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} \\
0 & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} \\
0 & 0 & u_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} \\
0 & 0 & 0 & u_{{4}{4}} & u_{{4}{5}} \\
0 & 0 & 0 & 0 & u_{{5}{5}} 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{ccccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} \\
l_{{2}{1}} & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} \\
l_{{3}{1}} & l_{{3}{2}} & u_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & u_{{4}{4}} & u_{{4}{5}} \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & u_{{5}{5}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{1}{1}} = u_{{1}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} u_{{1}{1}} $$
$$ a_{{3}{1}} = l_{{3}{1}} u_{{1}{1}} $$
$$ a_{{4}{1}} = l_{{4}{1}} u_{{1}{1}} $$
$$ a_{{5}{1}} = l_{{5}{1}} u_{{1}{1}} $$
$$ a_{{1}{2}} = u_{{1}{2}} $$
$$ a_{{2}{2}} = l_{{2}{1}} u_{{1}{2}} + u_{{2}{2}} $$
$$ a_{{3}{2}} = l_{{3}{1}} u_{{1}{2}} + l_{{3}{2}} u_{{2}{2}} $$
$$ a_{{4}{2}} = l_{{4}{1}} u_{{1}{2}} + l_{{4}{2}} u_{{2}{2}} $$
$$ a_{{5}{2}} = l_{{5}{1}} u_{{1}{2}} + l_{{5}{2}} u_{{2}{2}} $$
$$ a_{{1}{3}} = u_{{1}{3}} $$
$$ a_{{2}{3}} = l_{{2}{1}} u_{{1}{3}} + u_{{2}{3}} $$
$$ a_{{3}{3}} = l_{{3}{1}} u_{{1}{3}} + l_{{3}{2}} u_{{2}{3}} + u_{{3}{3}} $$
$$ a_{{4}{3}} = l_{{4}{1}} u_{{1}{3}} + l_{{4}{2}} u_{{2}{3}} + l_{{4}{3}} u_{{3}{3}} $$
$$ a_{{5}{3}} = l_{{5}{1}} u_{{1}{3}} + l_{{5}{2}} u_{{2}{3}} + l_{{5}{3}} u_{{3}{3}} $$
$$ a_{{1}{4}} = u_{{1}{4}} $$
$$ a_{{2}{4}} = l_{{2}{1}} u_{{1}{4}} + u_{{2}{4}} $$
$$ a_{{3}{4}} = l_{{3}{1}} u_{{1}{4}} + l_{{3}{2}} u_{{2}{4}} + u_{{3}{4}} $$
$$ a_{{4}{4}} = l_{{4}{1}} u_{{1}{4}} + l_{{4}{2}} u_{{2}{4}} + l_{{4}{3}} u_{{3}{4}} + u_{{4}{4}} $$
$$ a_{{5}{4}} = l_{{5}{1}} u_{{1}{4}} + l_{{5}{2}} u_{{2}{4}} + l_{{5}{3}} u_{{3}{4}} + l_{{5}{4}} u_{{4}{4}} $$
$$ a_{{1}{5}} = u_{{1}{5}} $$
$$ a_{{2}{5}} = l_{{2}{1}} u_{{1}{5}} + u_{{2}{5}} $$
$$ a_{{3}{5}} = l_{{3}{1}} u_{{1}{5}} + l_{{3}{2}} u_{{2}{5}} + u_{{3}{5}} $$
$$ a_{{4}{5}} = l_{{4}{1}} u_{{1}{5}} + l_{{4}{2}} u_{{2}{5}} + l_{{4}{3}} u_{{3}{5}} + u_{{4}{5}} $$
$$ a_{{5}{5}} = l_{{5}{1}} u_{{1}{5}} + l_{{5}{2}} u_{{2}{5}} + l_{{5}{3}} u_{{3}{5}} + l_{{5}{4}} u_{{4}{5}} + u_{{5}{5}} $$
\vfill\null
\columnbreak
$$ u_{{1}{1}} = a_{{1}{1}} $$
$$ l_{{2}{1}} = \dfrac{a_{{2}{1}}}{u_{{1}{1}}} $$
$$ l_{{3}{1}} = \dfrac{a_{{3}{1}}}{u_{{1}{1}}} $$
$$ l_{{4}{1}} = \dfrac{a_{{4}{1}}}{u_{{1}{1}}} $$
$$ l_{{5}{1}} = \dfrac{a_{{5}{1}}}{u_{{1}{1}}} $$
$$ u_{{1}{2}} = a_{{1}{2}} $$
$$ u_{{2}{2}} = a_{{2}{2}}- l_{{2}{1}} u_{{1}{2}} $$
$$ l_{{3}{2}} = \dfrac{a_{{3}{2}}- l_{{3}{1}} u_{{1}{2}}}{u_{{2}{2}}} $$
$$ l_{{4}{2}} = \dfrac{a_{{4}{2}}- l_{{4}{1}} u_{{1}{2}}}{u_{{2}{2}}} $$
$$ l_{{5}{2}} = \dfrac{a_{{5}{2}}- l_{{5}{1}} u_{{1}{2}}}{u_{{2}{2}}} $$
$$ u_{{1}{3}} = a_{{1}{3}} $$
$$ u_{{2}{3}} = a_{{2}{3}}- l_{{2}{1}} u_{{1}{3}} $$
$$ u_{{3}{3}} = a_{{3}{3}}- l_{{3}{1}} u_{{1}{3}}- l_{{3}{2}} u_{{2}{3}} $$
$$ l_{{4}{3}} = \dfrac{a_{{4}{3}}- l_{{4}{1}} u_{{1}{3}}- l_{{4}{2}} u_{{2}{3}}}{u_{{3}{3}}} $$
$$ l_{{5}{3}} = \dfrac{a_{{5}{3}}- l_{{5}{1}} u_{{1}{3}}- l_{{5}{2}} u_{{2}{3}}}{u_{{3}{3}}} $$
$$ u_{{1}{4}} = a_{{1}{4}} $$
$$ u_{{2}{4}} = a_{{2}{4}}- l_{{2}{1}} u_{{1}{4}} $$
$$ u_{{3}{4}} = a_{{3}{4}}- l_{{3}{1}} u_{{1}{4}}- l_{{3}{2}} u_{{2}{4}} $$
$$ u_{{4}{4}} = a_{{4}{4}}- l_{{4}{1}} u_{{1}{4}}- l_{{4}{2}} u_{{2}{4}}- l_{{4}{3}} u_{{3}{4}} $$
$$ l_{{5}{4}} = \dfrac{a_{{5}{4}}- l_{{5}{1}} u_{{1}{4}}- l_{{5}{2}} u_{{2}{4}}- l_{{5}{3}} u_{{3}{4}}}{u_{{4}{4}}} $$
$$ u_{{1}{5}} = a_{{1}{5}} $$
$$ u_{{2}{5}} = a_{{2}{5}}- l_{{2}{1}} u_{{1}{5}} $$
$$ u_{{3}{5}} = a_{{3}{5}}- l_{{3}{1}} u_{{1}{5}}- l_{{3}{2}} u_{{2}{5}} $$
$$ u_{{4}{5}} = a_{{4}{5}}- l_{{4}{1}} u_{{1}{5}}- l_{{4}{2}} u_{{2}{5}}- l_{{4}{3}} u_{{3}{5}} $$
$$ u_{{5}{5}} = a_{{5}{5}}- l_{{5}{1}} u_{{1}{5}}- l_{{5}{2}} u_{{2}{5}}- l_{{5}{3}} u_{{3}{5}}- l_{{5}{4}} u_{{4}{5}} $$
\end{multicols}
\subsection{Caso de matrices de $6\times 6$ }
$$ \left( 
\begin{array}{cccccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} & a_{{1}{4}} & a_{{1}{5}} & a_{{1}{6}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} & a_{{2}{4}} & a_{{2}{5}} & a_{{2}{6}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} & a_{{3}{4}} & a_{{3}{5}} & a_{{3}{6}} \\
a_{{4}{1}} & a_{{4}{2}} & a_{{4}{3}} & a_{{4}{4}} & a_{{4}{5}} & a_{{4}{6}} \\
a_{{5}{1}} & a_{{5}{2}} & a_{{5}{3}} & a_{{5}{4}} & a_{{5}{5}} & a_{{5}{6}} \\
a_{{6}{1}} & a_{{6}{2}} & a_{{6}{3}} & a_{{6}{4}} & a_{{6}{5}} & a_{{6}{6}} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0 \\
l_{{2}{1}} & 1 & 0 & 0 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & 1 & 0 & 0 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & 1 & 0 & 0 \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & 1 & 0 \\
l_{{6}{1}} & l_{{6}{2}} & l_{{6}{3}} & l_{{6}{4}} & l_{{6}{5}} & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{cccccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} & u_{{1}{6}} \\
0 & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} & u_{{2}{6}} \\
0 & 0 & u_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} & u_{{3}{6}} \\
0 & 0 & 0 & u_{{4}{4}} & u_{{4}{5}} & u_{{4}{6}} \\
0 & 0 & 0 & 0 & u_{{5}{5}} & u_{{5}{6}} \\
0 & 0 & 0 & 0 & 0 & u_{{6}{6}} 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{cccccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} & u_{{1}{6}} \\
l_{{2}{1}} & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} & u_{{2}{6}} \\
l_{{3}{1}} & l_{{3}{2}} & u_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} & u_{{3}{6}} \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & u_{{4}{4}} & u_{{4}{5}} & u_{{4}{6}} \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & u_{{5}{5}} & u_{{5}{6}} \\
l_{{6}{1}} & l_{{6}{2}} & l_{{6}{3}} & l_{{6}{4}} & l_{{6}{5}} & u_{{6}{6}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{1}{1}} = u_{{1}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} u_{{1}{1}} $$
$$ a_{{3}{1}} = l_{{3}{1}} u_{{1}{1}} $$
$$ a_{{4}{1}} = l_{{4}{1}} u_{{1}{1}} $$
$$ a_{{5}{1}} = l_{{5}{1}} u_{{1}{1}} $$
$$ a_{{6}{1}} = l_{{6}{1}} u_{{1}{1}} $$
$$ a_{{1}{2}} = u_{{1}{2}} $$
$$ a_{{2}{2}} = l_{{2}{1}} u_{{1}{2}} + u_{{2}{2}} $$
$$ a_{{3}{2}} = l_{{3}{1}} u_{{1}{2}} + l_{{3}{2}} u_{{2}{2}} $$
$$ a_{{4}{2}} = l_{{4}{1}} u_{{1}{2}} + l_{{4}{2}} u_{{2}{2}} $$
$$ a_{{5}{2}} = l_{{5}{1}} u_{{1}{2}} + l_{{5}{2}} u_{{2}{2}} $$
$$ a_{{6}{2}} = l_{{6}{1}} u_{{1}{2}} + l_{{6}{2}} u_{{2}{2}} $$
$$ a_{{1}{3}} = u_{{1}{3}} $$
$$ a_{{2}{3}} = l_{{2}{1}} u_{{1}{3}} + u_{{2}{3}} $$
$$ a_{{3}{3}} = l_{{3}{1}} u_{{1}{3}} + l_{{3}{2}} u_{{2}{3}} + u_{{3}{3}} $$
$$ a_{{4}{3}} = l_{{4}{1}} u_{{1}{3}} + l_{{4}{2}} u_{{2}{3}} + l_{{4}{3}} u_{{3}{3}} $$
$$ a_{{5}{3}} = l_{{5}{1}} u_{{1}{3}} + l_{{5}{2}} u_{{2}{3}} + l_{{5}{3}} u_{{3}{3}} $$
$$ a_{{6}{3}} = l_{{6}{1}} u_{{1}{3}} + l_{{6}{2}} u_{{2}{3}} + l_{{6}{3}} u_{{3}{3}} $$
$$ a_{{1}{4}} = u_{{1}{4}} $$
$$ a_{{2}{4}} = l_{{2}{1}} u_{{1}{4}} + u_{{2}{4}} $$
$$ a_{{3}{4}} = l_{{3}{1}} u_{{1}{4}} + l_{{3}{2}} u_{{2}{4}} + u_{{3}{4}} $$
$$ a_{{4}{4}} = l_{{4}{1}} u_{{1}{4}} + l_{{4}{2}} u_{{2}{4}} + l_{{4}{3}} u_{{3}{4}} + u_{{4}{4}} $$
$$ a_{{5}{4}} = l_{{5}{1}} u_{{1}{4}} + l_{{5}{2}} u_{{2}{4}} + l_{{5}{3}} u_{{3}{4}} + l_{{5}{4}} u_{{4}{4}} $$
$$ a_{{6}{4}} = l_{{6}{1}} u_{{1}{4}} + l_{{6}{2}} u_{{2}{4}} + l_{{6}{3}} u_{{3}{4}} + l_{{6}{4}} u_{{4}{4}} $$
$$ a_{{1}{5}} = u_{{1}{5}} $$
$$ a_{{2}{5}} = l_{{2}{1}} u_{{1}{5}} + u_{{2}{5}} $$
$$ a_{{3}{5}} = l_{{3}{1}} u_{{1}{5}} + l_{{3}{2}} u_{{2}{5}} + u_{{3}{5}} $$
$$ a_{{4}{5}} = l_{{4}{1}} u_{{1}{5}} + l_{{4}{2}} u_{{2}{5}} + l_{{4}{3}} u_{{3}{5}} + u_{{4}{5}} $$
$$ a_{{5}{5}} = l_{{5}{1}} u_{{1}{5}} + l_{{5}{2}} u_{{2}{5}} + l_{{5}{3}} u_{{3}{5}} + l_{{5}{4}} u_{{4}{5}} + u_{{5}{5}} $$
$$ a_{{6}{5}} = l_{{6}{1}} u_{{1}{5}} + l_{{6}{2}} u_{{2}{5}} + l_{{6}{3}} u_{{3}{5}} + l_{{6}{4}} u_{{4}{5}} + l_{{6}{5}} u_{{5}{5}} $$
$$ a_{{1}{6}} = u_{{1}{6}} $$
$$ a_{{2}{6}} = l_{{2}{1}} u_{{1}{6}} + u_{{2}{6}} $$
$$ a_{{3}{6}} = l_{{3}{1}} u_{{1}{6}} + l_{{3}{2}} u_{{2}{6}} + u_{{3}{6}} $$
$$ a_{{4}{6}} = l_{{4}{1}} u_{{1}{6}} + l_{{4}{2}} u_{{2}{6}} + l_{{4}{3}} u_{{3}{6}} + u_{{4}{6}} $$
$$ a_{{5}{6}} = l_{{5}{1}} u_{{1}{6}} + l_{{5}{2}} u_{{2}{6}} + l_{{5}{3}} u_{{3}{6}} + l_{{5}{4}} u_{{4}{6}} + u_{{5}{6}} $$
$$ a_{{6}{6}} = l_{{6}{1}} u_{{1}{6}} + l_{{6}{2}} u_{{2}{6}} + l_{{6}{3}} u_{{3}{6}} + l_{{6}{4}} u_{{4}{6}} + l_{{6}{5}} u_{{5}{6}} + u_{{6}{6}} $$
\vfill\null
\columnbreak
$$ u_{{1}{1}} = a_{{1}{1}} $$
$$ l_{{2}{1}} = \dfrac{a_{{2}{1}}}{u_{{1}{1}}} $$
$$ l_{{3}{1}} = \dfrac{a_{{3}{1}}}{u_{{1}{1}}} $$
$$ l_{{4}{1}} = \dfrac{a_{{4}{1}}}{u_{{1}{1}}} $$
$$ l_{{5}{1}} = \dfrac{a_{{5}{1}}}{u_{{1}{1}}} $$
$$ l_{{6}{1}} = \dfrac{a_{{6}{1}}}{u_{{1}{1}}} $$
$$ u_{{1}{2}} = a_{{1}{2}} $$
$$ u_{{2}{2}} = a_{{2}{2}}- l_{{2}{1}} u_{{1}{2}} $$
$$ l_{{3}{2}} = \dfrac{a_{{3}{2}}- l_{{3}{1}} u_{{1}{2}}}{u_{{2}{2}}} $$
$$ l_{{4}{2}} = \dfrac{a_{{4}{2}}- l_{{4}{1}} u_{{1}{2}}}{u_{{2}{2}}} $$
$$ l_{{5}{2}} = \dfrac{a_{{5}{2}}- l_{{5}{1}} u_{{1}{2}}}{u_{{2}{2}}} $$
$$ l_{{6}{2}} = \dfrac{a_{{6}{2}}- l_{{6}{1}} u_{{1}{2}}}{u_{{2}{2}}} $$
$$ u_{{1}{3}} = a_{{1}{3}} $$
$$ u_{{2}{3}} = a_{{2}{3}}- l_{{2}{1}} u_{{1}{3}} $$
$$ u_{{3}{3}} = a_{{3}{3}}- l_{{3}{1}} u_{{1}{3}}- l_{{3}{2}} u_{{2}{3}} $$
$$ l_{{4}{3}} = \dfrac{a_{{4}{3}}- l_{{4}{1}} u_{{1}{3}}- l_{{4}{2}} u_{{2}{3}}}{u_{{3}{3}}} $$
$$ l_{{5}{3}} = \dfrac{a_{{5}{3}}- l_{{5}{1}} u_{{1}{3}}- l_{{5}{2}} u_{{2}{3}}}{u_{{3}{3}}} $$
$$ l_{{6}{3}} = \dfrac{a_{{6}{3}}- l_{{6}{1}} u_{{1}{3}}- l_{{6}{2}} u_{{2}{3}}}{u_{{3}{3}}} $$
$$ u_{{1}{4}} = a_{{1}{4}} $$
$$ u_{{2}{4}} = a_{{2}{4}}- l_{{2}{1}} u_{{1}{4}} $$
$$ u_{{3}{4}} = a_{{3}{4}}- l_{{3}{1}} u_{{1}{4}}- l_{{3}{2}} u_{{2}{4}} $$
$$ u_{{4}{4}} = a_{{4}{4}}- l_{{4}{1}} u_{{1}{4}}- l_{{4}{2}} u_{{2}{4}}- l_{{4}{3}} u_{{3}{4}} $$
$$ l_{{5}{4}} = \dfrac{a_{{5}{4}}- l_{{5}{1}} u_{{1}{4}}- l_{{5}{2}} u_{{2}{4}}- l_{{5}{3}} u_{{3}{4}}}{u_{{4}{4}}} $$
$$ l_{{6}{4}} = \dfrac{a_{{6}{4}}- l_{{6}{1}} u_{{1}{4}}- l_{{6}{2}} u_{{2}{4}}- l_{{6}{3}} u_{{3}{4}}}{u_{{4}{4}}} $$
$$ u_{{1}{5}} = a_{{1}{5}} $$
$$ u_{{2}{5}} = a_{{2}{5}}- l_{{2}{1}} u_{{1}{5}} $$
$$ u_{{3}{5}} = a_{{3}{5}}- l_{{3}{1}} u_{{1}{5}}- l_{{3}{2}} u_{{2}{5}} $$
$$ u_{{4}{5}} = a_{{4}{5}}- l_{{4}{1}} u_{{1}{5}}- l_{{4}{2}} u_{{2}{5}}- l_{{4}{3}} u_{{3}{5}} $$
$$ u_{{5}{5}} = a_{{5}{5}}- l_{{5}{1}} u_{{1}{5}}- l_{{5}{2}} u_{{2}{5}}- l_{{5}{3}} u_{{3}{5}}- l_{{5}{4}} u_{{4}{5}} $$
$$ l_{{6}{5}} = \dfrac{a_{{6}{5}}- l_{{6}{1}} u_{{1}{5}}- l_{{6}{2}} u_{{2}{5}}- l_{{6}{3}} u_{{3}{5}}- l_{{6}{4}} u_{{4}{5}}}{u_{{5}{5}}} $$
$$ u_{{1}{6}} = a_{{1}{6}} $$
$$ u_{{2}{6}} = a_{{2}{6}}- l_{{2}{1}} u_{{1}{6}} $$
$$ u_{{3}{6}} = a_{{3}{6}}- l_{{3}{1}} u_{{1}{6}}- l_{{3}{2}} u_{{2}{6}} $$
$$ u_{{4}{6}} = a_{{4}{6}}- l_{{4}{1}} u_{{1}{6}}- l_{{4}{2}} u_{{2}{6}}- l_{{4}{3}} u_{{3}{6}} $$
$$ u_{{5}{6}} = a_{{5}{6}}- l_{{5}{1}} u_{{1}{6}}- l_{{5}{2}} u_{{2}{6}}- l_{{5}{3}} u_{{3}{6}}- l_{{5}{4}} u_{{4}{6}} $$
$$ u_{{6}{6}} = a_{{6}{6}}- l_{{6}{1}} u_{{1}{6}}- l_{{6}{2}} u_{{2}{6}}- l_{{6}{3}} u_{{3}{6}}- l_{{6}{4}} u_{{4}{6}}- l_{{6}{5}} u_{{5}{6}} $$
\end{multicols}
\section{Metodo uL}
\subsection{Caso de matrices de $2\times 2$ }
$$ \left( 
\begin{array}{cc}
a_{{1}{1}} & a_{{1}{2}} \\
a_{{2}{1}} & a_{{2}{2}} 
 \end{array}
\right)
 = \left( 
\begin{array}{cc}
1 & u_{{1}{2}} \\
0 & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{cc}
l_{{1}{1}} & 0 \\
l_{{2}{1}} & l_{{2}{2}} 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{cc}
l_{{1}{1}} & u_{{1}{2}} \\
l_{{2}{1}} & l_{{2}{2}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{2}{2}} = l_{{2}{2}} $$
$$ a_{{1}{2}} = u_{{1}{2}} l_{{2}{2}} $$
$$ a_{{2}{1}} = l_{{2}{1}} $$
$$ a_{{1}{1}} = l_{{1}{1}} + u_{{1}{2}} l_{{2}{1}} $$
\vfill\null
\columnbreak
$$ l_{{2}{2}} = a_{{2}{2}} $$
$$ u_{{1}{2}} = \dfrac{a_{{1}{2}}}{l_{{2}{2}}} $$
$$ l_{{2}{1}} = a_{{2}{1}} $$
$$ l_{{1}{1}} = a_{{1}{1}}- u_{{1}{2}} l_{{2}{1}} $$
\end{multicols}
\subsection{Caso de matrices de $3\times 3$ }
$$ \left( 
\begin{array}{ccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccc}
1 & u_{{1}{2}} & u_{{1}{3}} \\
0 & 1 & u_{{2}{3}} \\
0 & 0 & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{ccc}
l_{{1}{1}} & 0 & 0 \\
l_{{2}{1}} & l_{{2}{2}} & 0 \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{ccc}
l_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} \\
l_{{2}{1}} & l_{{2}{2}} & u_{{2}{3}} \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{3}{3}} = l_{{3}{3}} $$
$$ a_{{2}{3}} = u_{{2}{3}} l_{{3}{3}} $$
$$ a_{{1}{3}} = u_{{1}{3}} l_{{3}{3}} $$
$$ a_{{3}{2}} = l_{{3}{2}} $$
$$ a_{{2}{2}} = l_{{2}{2}} + u_{{2}{3}} l_{{3}{2}} $$
$$ a_{{1}{2}} = u_{{1}{2}} l_{{2}{2}} + u_{{1}{3}} l_{{3}{2}} $$
$$ a_{{3}{1}} = l_{{3}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} + u_{{2}{3}} l_{{3}{1}} $$
$$ a_{{1}{1}} = l_{{1}{1}} + u_{{1}{2}} l_{{2}{1}} + u_{{1}{3}} l_{{3}{1}} $$
\vfill\null
\columnbreak
$$ l_{{3}{3}} = a_{{3}{3}} $$
$$ u_{{2}{3}} = \dfrac{a_{{2}{3}}}{l_{{3}{3}}} $$
$$ u_{{1}{3}} = \dfrac{a_{{1}{3}}}{l_{{3}{3}}} $$
$$ l_{{3}{2}} = a_{{3}{2}} $$
$$ l_{{2}{2}} = a_{{2}{2}}- u_{{2}{3}} l_{{3}{2}} $$
$$ u_{{1}{2}} = \dfrac{a_{{1}{2}}- u_{{1}{3}} l_{{3}{2}}}{l_{{2}{2}}} $$
$$ l_{{3}{1}} = a_{{3}{1}} $$
$$ l_{{2}{1}} = a_{{2}{1}}- u_{{2}{3}} l_{{3}{1}} $$
$$ l_{{1}{1}} = a_{{1}{1}}- u_{{1}{2}} l_{{2}{1}}- u_{{1}{3}} l_{{3}{1}} $$
\end{multicols}
\subsection{Caso de matrices de $4\times 4$ }
$$ \left( 
\begin{array}{cccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} & a_{{1}{4}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} & a_{{2}{4}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} & a_{{3}{4}} \\
a_{{4}{1}} & a_{{4}{2}} & a_{{4}{3}} & a_{{4}{4}} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccc}
1 & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} \\
0 & 1 & u_{{2}{3}} & u_{{2}{4}} \\
0 & 0 & 1 & u_{{3}{4}} \\
0 & 0 & 0 & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{cccc}
l_{{1}{1}} & 0 & 0 & 0 \\
l_{{2}{1}} & l_{{2}{2}} & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{cccc}
l_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} \\
l_{{2}{1}} & l_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & u_{{3}{4}} \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{4}{4}} = l_{{4}{4}} $$
$$ a_{{3}{4}} = u_{{3}{4}} l_{{4}{4}} $$
$$ a_{{2}{4}} = u_{{2}{4}} l_{{4}{4}} $$
$$ a_{{1}{4}} = u_{{1}{4}} l_{{4}{4}} $$
$$ a_{{4}{3}} = l_{{4}{3}} $$
$$ a_{{3}{3}} = l_{{3}{3}} + u_{{3}{4}} l_{{4}{3}} $$
$$ a_{{2}{3}} = u_{{2}{3}} l_{{3}{3}} + u_{{2}{4}} l_{{4}{3}} $$
$$ a_{{1}{3}} = u_{{1}{3}} l_{{3}{3}} + u_{{1}{4}} l_{{4}{3}} $$
$$ a_{{4}{2}} = l_{{4}{2}} $$
$$ a_{{3}{2}} = l_{{3}{2}} + u_{{3}{4}} l_{{4}{2}} $$
$$ a_{{2}{2}} = l_{{2}{2}} + u_{{2}{3}} l_{{3}{2}} + u_{{2}{4}} l_{{4}{2}} $$
$$ a_{{1}{2}} = u_{{1}{2}} l_{{2}{2}} + u_{{1}{3}} l_{{3}{2}} + u_{{1}{4}} l_{{4}{2}} $$
$$ a_{{4}{1}} = l_{{4}{1}} $$
$$ a_{{3}{1}} = l_{{3}{1}} + u_{{3}{4}} l_{{4}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} + u_{{2}{3}} l_{{3}{1}} + u_{{2}{4}} l_{{4}{1}} $$
$$ a_{{1}{1}} = l_{{1}{1}} + u_{{1}{2}} l_{{2}{1}} + u_{{1}{3}} l_{{3}{1}} + u_{{1}{4}} l_{{4}{1}} $$
\vfill\null
\columnbreak
$$ l_{{4}{4}} = a_{{4}{4}} $$
$$ u_{{3}{4}} = \dfrac{a_{{3}{4}}}{l_{{4}{4}}} $$
$$ u_{{2}{4}} = \dfrac{a_{{2}{4}}}{l_{{4}{4}}} $$
$$ u_{{1}{4}} = \dfrac{a_{{1}{4}}}{l_{{4}{4}}} $$
$$ l_{{4}{3}} = a_{{4}{3}} $$
$$ l_{{3}{3}} = a_{{3}{3}}- u_{{3}{4}} l_{{4}{3}} $$
$$ u_{{2}{3}} = \dfrac{a_{{2}{3}}- u_{{2}{4}} l_{{4}{3}}}{l_{{3}{3}}} $$
$$ u_{{1}{3}} = \dfrac{a_{{1}{3}}- u_{{1}{4}} l_{{4}{3}}}{l_{{3}{3}}} $$
$$ l_{{4}{2}} = a_{{4}{2}} $$
$$ l_{{3}{2}} = a_{{3}{2}}- u_{{3}{4}} l_{{4}{2}} $$
$$ l_{{2}{2}} = a_{{2}{2}}- u_{{2}{3}} l_{{3}{2}}- u_{{2}{4}} l_{{4}{2}} $$
$$ u_{{1}{2}} = \dfrac{a_{{1}{2}}- u_{{1}{3}} l_{{3}{2}}- u_{{1}{4}} l_{{4}{2}}}{l_{{2}{2}}} $$
$$ l_{{4}{1}} = a_{{4}{1}} $$
$$ l_{{3}{1}} = a_{{3}{1}}- u_{{3}{4}} l_{{4}{1}} $$
$$ l_{{2}{1}} = a_{{2}{1}}- u_{{2}{3}} l_{{3}{1}}- u_{{2}{4}} l_{{4}{1}} $$
$$ l_{{1}{1}} = a_{{1}{1}}- u_{{1}{2}} l_{{2}{1}}- u_{{1}{3}} l_{{3}{1}}- u_{{1}{4}} l_{{4}{1}} $$
\end{multicols}
\subsection{Caso de matrices de $5\times 5$ }
$$ \left( 
\begin{array}{ccccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} & a_{{1}{4}} & a_{{1}{5}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} & a_{{2}{4}} & a_{{2}{5}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} & a_{{3}{4}} & a_{{3}{5}} \\
a_{{4}{1}} & a_{{4}{2}} & a_{{4}{3}} & a_{{4}{4}} & a_{{4}{5}} \\
a_{{5}{1}} & a_{{5}{2}} & a_{{5}{3}} & a_{{5}{4}} & a_{{5}{5}} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccccc}
1 & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} \\
0 & 1 & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} \\
0 & 0 & 1 & u_{{3}{4}} & u_{{3}{5}} \\
0 & 0 & 0 & 1 & u_{{4}{5}} \\
0 & 0 & 0 & 0 & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{ccccc}
l_{{1}{1}} & 0 & 0 & 0 & 0 \\
l_{{2}{1}} & l_{{2}{2}} & 0 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & 0 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} & 0 \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & l_{{5}{5}} 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{ccccc}
l_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} \\
l_{{2}{1}} & l_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} & u_{{4}{5}} \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & l_{{5}{5}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{5}{5}} = l_{{5}{5}} $$
$$ a_{{4}{5}} = u_{{4}{5}} l_{{5}{5}} $$
$$ a_{{3}{5}} = u_{{3}{5}} l_{{5}{5}} $$
$$ a_{{2}{5}} = u_{{2}{5}} l_{{5}{5}} $$
$$ a_{{1}{5}} = u_{{1}{5}} l_{{5}{5}} $$
$$ a_{{5}{4}} = l_{{5}{4}} $$
$$ a_{{4}{4}} = l_{{4}{4}} + u_{{4}{5}} l_{{5}{4}} $$
$$ a_{{3}{4}} = u_{{3}{4}} l_{{4}{4}} + u_{{3}{5}} l_{{5}{4}} $$
$$ a_{{2}{4}} = u_{{2}{4}} l_{{4}{4}} + u_{{2}{5}} l_{{5}{4}} $$
$$ a_{{1}{4}} = u_{{1}{4}} l_{{4}{4}} + u_{{1}{5}} l_{{5}{4}} $$
$$ a_{{5}{3}} = l_{{5}{3}} $$
$$ a_{{4}{3}} = l_{{4}{3}} + u_{{4}{5}} l_{{5}{3}} $$
$$ a_{{3}{3}} = l_{{3}{3}} + u_{{3}{4}} l_{{4}{3}} + u_{{3}{5}} l_{{5}{3}} $$
$$ a_{{2}{3}} = u_{{2}{3}} l_{{3}{3}} + u_{{2}{4}} l_{{4}{3}} + u_{{2}{5}} l_{{5}{3}} $$
$$ a_{{1}{3}} = u_{{1}{3}} l_{{3}{3}} + u_{{1}{4}} l_{{4}{3}} + u_{{1}{5}} l_{{5}{3}} $$
$$ a_{{5}{2}} = l_{{5}{2}} $$
$$ a_{{4}{2}} = l_{{4}{2}} + u_{{4}{5}} l_{{5}{2}} $$
$$ a_{{3}{2}} = l_{{3}{2}} + u_{{3}{4}} l_{{4}{2}} + u_{{3}{5}} l_{{5}{2}} $$
$$ a_{{2}{2}} = l_{{2}{2}} + u_{{2}{3}} l_{{3}{2}} + u_{{2}{4}} l_{{4}{2}} + u_{{2}{5}} l_{{5}{2}} $$
$$ a_{{1}{2}} = u_{{1}{2}} l_{{2}{2}} + u_{{1}{3}} l_{{3}{2}} + u_{{1}{4}} l_{{4}{2}} + u_{{1}{5}} l_{{5}{2}} $$
$$ a_{{5}{1}} = l_{{5}{1}} $$
$$ a_{{4}{1}} = l_{{4}{1}} + u_{{4}{5}} l_{{5}{1}} $$
$$ a_{{3}{1}} = l_{{3}{1}} + u_{{3}{4}} l_{{4}{1}} + u_{{3}{5}} l_{{5}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} + u_{{2}{3}} l_{{3}{1}} + u_{{2}{4}} l_{{4}{1}} + u_{{2}{5}} l_{{5}{1}} $$
$$ a_{{1}{1}} = l_{{1}{1}} + u_{{1}{2}} l_{{2}{1}} + u_{{1}{3}} l_{{3}{1}} + u_{{1}{4}} l_{{4}{1}} + u_{{1}{5}} l_{{5}{1}} $$
\vfill\null
\columnbreak
$$ l_{{5}{5}} = a_{{5}{5}} $$
$$ u_{{4}{5}} = \dfrac{a_{{4}{5}}}{l_{{5}{5}}} $$
$$ u_{{3}{5}} = \dfrac{a_{{3}{5}}}{l_{{5}{5}}} $$
$$ u_{{2}{5}} = \dfrac{a_{{2}{5}}}{l_{{5}{5}}} $$
$$ u_{{1}{5}} = \dfrac{a_{{1}{5}}}{l_{{5}{5}}} $$
$$ l_{{5}{4}} = a_{{5}{4}} $$
$$ l_{{4}{4}} = a_{{4}{4}}- u_{{4}{5}} l_{{5}{4}} $$
$$ u_{{3}{4}} = \dfrac{a_{{3}{4}}- u_{{3}{5}} l_{{5}{4}}}{l_{{4}{4}}} $$
$$ u_{{2}{4}} = \dfrac{a_{{2}{4}}- u_{{2}{5}} l_{{5}{4}}}{l_{{4}{4}}} $$
$$ u_{{1}{4}} = \dfrac{a_{{1}{4}}- u_{{1}{5}} l_{{5}{4}}}{l_{{4}{4}}} $$
$$ l_{{5}{3}} = a_{{5}{3}} $$
$$ l_{{4}{3}} = a_{{4}{3}}- u_{{4}{5}} l_{{5}{3}} $$
$$ l_{{3}{3}} = a_{{3}{3}}- u_{{3}{4}} l_{{4}{3}}- u_{{3}{5}} l_{{5}{3}} $$
$$ u_{{2}{3}} = \dfrac{a_{{2}{3}}- u_{{2}{4}} l_{{4}{3}}- u_{{2}{5}} l_{{5}{3}}}{l_{{3}{3}}} $$
$$ u_{{1}{3}} = \dfrac{a_{{1}{3}}- u_{{1}{4}} l_{{4}{3}}- u_{{1}{5}} l_{{5}{3}}}{l_{{3}{3}}} $$
$$ l_{{5}{2}} = a_{{5}{2}} $$
$$ l_{{4}{2}} = a_{{4}{2}}- u_{{4}{5}} l_{{5}{2}} $$
$$ l_{{3}{2}} = a_{{3}{2}}- u_{{3}{4}} l_{{4}{2}}- u_{{3}{5}} l_{{5}{2}} $$
$$ l_{{2}{2}} = a_{{2}{2}}- u_{{2}{3}} l_{{3}{2}}- u_{{2}{4}} l_{{4}{2}}- u_{{2}{5}} l_{{5}{2}} $$
$$ u_{{1}{2}} = \dfrac{a_{{1}{2}}- u_{{1}{3}} l_{{3}{2}}- u_{{1}{4}} l_{{4}{2}}- u_{{1}{5}} l_{{5}{2}}}{l_{{2}{2}}} $$
$$ l_{{5}{1}} = a_{{5}{1}} $$
$$ l_{{4}{1}} = a_{{4}{1}}- u_{{4}{5}} l_{{5}{1}} $$
$$ l_{{3}{1}} = a_{{3}{1}}- u_{{3}{4}} l_{{4}{1}}- u_{{3}{5}} l_{{5}{1}} $$
$$ l_{{2}{1}} = a_{{2}{1}}- u_{{2}{3}} l_{{3}{1}}- u_{{2}{4}} l_{{4}{1}}- u_{{2}{5}} l_{{5}{1}} $$
$$ l_{{1}{1}} = a_{{1}{1}}- u_{{1}{2}} l_{{2}{1}}- u_{{1}{3}} l_{{3}{1}}- u_{{1}{4}} l_{{4}{1}}- u_{{1}{5}} l_{{5}{1}} $$
\end{multicols}
\subsection{Caso de matrices de $6\times 6$ }
$$ \left( 
\begin{array}{cccccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} & a_{{1}{4}} & a_{{1}{5}} & a_{{1}{6}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} & a_{{2}{4}} & a_{{2}{5}} & a_{{2}{6}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} & a_{{3}{4}} & a_{{3}{5}} & a_{{3}{6}} \\
a_{{4}{1}} & a_{{4}{2}} & a_{{4}{3}} & a_{{4}{4}} & a_{{4}{5}} & a_{{4}{6}} \\
a_{{5}{1}} & a_{{5}{2}} & a_{{5}{3}} & a_{{5}{4}} & a_{{5}{5}} & a_{{5}{6}} \\
a_{{6}{1}} & a_{{6}{2}} & a_{{6}{3}} & a_{{6}{4}} & a_{{6}{5}} & a_{{6}{6}} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccccc}
1 & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} & u_{{1}{6}} \\
0 & 1 & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} & u_{{2}{6}} \\
0 & 0 & 1 & u_{{3}{4}} & u_{{3}{5}} & u_{{3}{6}} \\
0 & 0 & 0 & 1 & u_{{4}{5}} & u_{{4}{6}} \\
0 & 0 & 0 & 0 & 1 & u_{{5}{6}} \\
0 & 0 & 0 & 0 & 0 & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{cccccc}
l_{{1}{1}} & 0 & 0 & 0 & 0 & 0 \\
l_{{2}{1}} & l_{{2}{2}} & 0 & 0 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & 0 & 0 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} & 0 & 0 \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & l_{{5}{5}} & 0 \\
l_{{6}{1}} & l_{{6}{2}} & l_{{6}{3}} & l_{{6}{4}} & l_{{6}{5}} & l_{{6}{6}} 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{cccccc}
l_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} & u_{{1}{6}} \\
l_{{2}{1}} & l_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} & u_{{2}{6}} \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} & u_{{3}{6}} \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} & u_{{4}{5}} & u_{{4}{6}} \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & l_{{5}{5}} & u_{{5}{6}} \\
l_{{6}{1}} & l_{{6}{2}} & l_{{6}{3}} & l_{{6}{4}} & l_{{6}{5}} & l_{{6}{6}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{6}{6}} = l_{{6}{6}} $$
$$ a_{{5}{6}} = u_{{5}{6}} l_{{6}{6}} $$
$$ a_{{4}{6}} = u_{{4}{6}} l_{{6}{6}} $$
$$ a_{{3}{6}} = u_{{3}{6}} l_{{6}{6}} $$
$$ a_{{2}{6}} = u_{{2}{6}} l_{{6}{6}} $$
$$ a_{{1}{6}} = u_{{1}{6}} l_{{6}{6}} $$
$$ a_{{6}{5}} = l_{{6}{5}} $$
$$ a_{{5}{5}} = l_{{5}{5}} + u_{{5}{6}} l_{{6}{5}} $$
$$ a_{{4}{5}} = u_{{4}{5}} l_{{5}{5}} + u_{{4}{6}} l_{{6}{5}} $$
$$ a_{{3}{5}} = u_{{3}{5}} l_{{5}{5}} + u_{{3}{6}} l_{{6}{5}} $$
$$ a_{{2}{5}} = u_{{2}{5}} l_{{5}{5}} + u_{{2}{6}} l_{{6}{5}} $$
$$ a_{{1}{5}} = u_{{1}{5}} l_{{5}{5}} + u_{{1}{6}} l_{{6}{5}} $$
$$ a_{{6}{4}} = l_{{6}{4}} $$
$$ a_{{5}{4}} = l_{{5}{4}} + u_{{5}{6}} l_{{6}{4}} $$
$$ a_{{4}{4}} = l_{{4}{4}} + u_{{4}{5}} l_{{5}{4}} + u_{{4}{6}} l_{{6}{4}} $$
$$ a_{{3}{4}} = u_{{3}{4}} l_{{4}{4}} + u_{{3}{5}} l_{{5}{4}} + u_{{3}{6}} l_{{6}{4}} $$
$$ a_{{2}{4}} = u_{{2}{4}} l_{{4}{4}} + u_{{2}{5}} l_{{5}{4}} + u_{{2}{6}} l_{{6}{4}} $$
$$ a_{{1}{4}} = u_{{1}{4}} l_{{4}{4}} + u_{{1}{5}} l_{{5}{4}} + u_{{1}{6}} l_{{6}{4}} $$
$$ a_{{6}{3}} = l_{{6}{3}} $$
$$ a_{{5}{3}} = l_{{5}{3}} + u_{{5}{6}} l_{{6}{3}} $$
$$ a_{{4}{3}} = l_{{4}{3}} + u_{{4}{5}} l_{{5}{3}} + u_{{4}{6}} l_{{6}{3}} $$
$$ a_{{3}{3}} = l_{{3}{3}} + u_{{3}{4}} l_{{4}{3}} + u_{{3}{5}} l_{{5}{3}} + u_{{3}{6}} l_{{6}{3}} $$
$$ a_{{2}{3}} = u_{{2}{3}} l_{{3}{3}} + u_{{2}{4}} l_{{4}{3}} + u_{{2}{5}} l_{{5}{3}} + u_{{2}{6}} l_{{6}{3}} $$
$$ a_{{1}{3}} = u_{{1}{3}} l_{{3}{3}} + u_{{1}{4}} l_{{4}{3}} + u_{{1}{5}} l_{{5}{3}} + u_{{1}{6}} l_{{6}{3}} $$
$$ a_{{6}{2}} = l_{{6}{2}} $$
$$ a_{{5}{2}} = l_{{5}{2}} + u_{{5}{6}} l_{{6}{2}} $$
$$ a_{{4}{2}} = l_{{4}{2}} + u_{{4}{5}} l_{{5}{2}} + u_{{4}{6}} l_{{6}{2}} $$
$$ a_{{3}{2}} = l_{{3}{2}} + u_{{3}{4}} l_{{4}{2}} + u_{{3}{5}} l_{{5}{2}} + u_{{3}{6}} l_{{6}{2}} $$
$$ a_{{2}{2}} = l_{{2}{2}} + u_{{2}{3}} l_{{3}{2}} + u_{{2}{4}} l_{{4}{2}} + u_{{2}{5}} l_{{5}{2}} + u_{{2}{6}} l_{{6}{2}} $$
$$ a_{{1}{2}} = u_{{1}{2}} l_{{2}{2}} + u_{{1}{3}} l_{{3}{2}} + u_{{1}{4}} l_{{4}{2}} + u_{{1}{5}} l_{{5}{2}} + u_{{1}{6}} l_{{6}{2}} $$
$$ a_{{6}{1}} = l_{{6}{1}} $$
$$ a_{{5}{1}} = l_{{5}{1}} + u_{{5}{6}} l_{{6}{1}} $$
$$ a_{{4}{1}} = l_{{4}{1}} + u_{{4}{5}} l_{{5}{1}} + u_{{4}{6}} l_{{6}{1}} $$
$$ a_{{3}{1}} = l_{{3}{1}} + u_{{3}{4}} l_{{4}{1}} + u_{{3}{5}} l_{{5}{1}} + u_{{3}{6}} l_{{6}{1}} $$
$$ a_{{2}{1}} = l_{{2}{1}} + u_{{2}{3}} l_{{3}{1}} + u_{{2}{4}} l_{{4}{1}} + u_{{2}{5}} l_{{5}{1}} + u_{{2}{6}} l_{{6}{1}} $$
$$ a_{{1}{1}} = l_{{1}{1}} + u_{{1}{2}} l_{{2}{1}} + u_{{1}{3}} l_{{3}{1}} + u_{{1}{4}} l_{{4}{1}} + u_{{1}{5}} l_{{5}{1}} + u_{{1}{6}} l_{{6}{1}} $$
\vfill\null
\columnbreak
$$ l_{{6}{6}} = a_{{6}{6}} $$
$$ u_{{5}{6}} = \dfrac{a_{{5}{6}}}{l_{{6}{6}}} $$
$$ u_{{4}{6}} = \dfrac{a_{{4}{6}}}{l_{{6}{6}}} $$
$$ u_{{3}{6}} = \dfrac{a_{{3}{6}}}{l_{{6}{6}}} $$
$$ u_{{2}{6}} = \dfrac{a_{{2}{6}}}{l_{{6}{6}}} $$
$$ u_{{1}{6}} = \dfrac{a_{{1}{6}}}{l_{{6}{6}}} $$
$$ l_{{6}{5}} = a_{{6}{5}} $$
$$ l_{{5}{5}} = a_{{5}{5}}- u_{{5}{6}} l_{{6}{5}} $$
$$ u_{{4}{5}} = \dfrac{a_{{4}{5}}- u_{{4}{6}} l_{{6}{5}}}{l_{{5}{5}}} $$
$$ u_{{3}{5}} = \dfrac{a_{{3}{5}}- u_{{3}{6}} l_{{6}{5}}}{l_{{5}{5}}} $$
$$ u_{{2}{5}} = \dfrac{a_{{2}{5}}- u_{{2}{6}} l_{{6}{5}}}{l_{{5}{5}}} $$
$$ u_{{1}{5}} = \dfrac{a_{{1}{5}}- u_{{1}{6}} l_{{6}{5}}}{l_{{5}{5}}} $$
$$ l_{{6}{4}} = a_{{6}{4}} $$
$$ l_{{5}{4}} = a_{{5}{4}}- u_{{5}{6}} l_{{6}{4}} $$
$$ l_{{4}{4}} = a_{{4}{4}}- u_{{4}{5}} l_{{5}{4}}- u_{{4}{6}} l_{{6}{4}} $$
$$ u_{{3}{4}} = \dfrac{a_{{3}{4}}- u_{{3}{5}} l_{{5}{4}}- u_{{3}{6}} l_{{6}{4}}}{l_{{4}{4}}} $$
$$ u_{{2}{4}} = \dfrac{a_{{2}{4}}- u_{{2}{5}} l_{{5}{4}}- u_{{2}{6}} l_{{6}{4}}}{l_{{4}{4}}} $$
$$ u_{{1}{4}} = \dfrac{a_{{1}{4}}- u_{{1}{5}} l_{{5}{4}}- u_{{1}{6}} l_{{6}{4}}}{l_{{4}{4}}} $$
$$ l_{{6}{3}} = a_{{6}{3}} $$
$$ l_{{5}{3}} = a_{{5}{3}}- u_{{5}{6}} l_{{6}{3}} $$
$$ l_{{4}{3}} = a_{{4}{3}}- u_{{4}{5}} l_{{5}{3}}- u_{{4}{6}} l_{{6}{3}} $$
$$ l_{{3}{3}} = a_{{3}{3}}- u_{{3}{4}} l_{{4}{3}}- u_{{3}{5}} l_{{5}{3}}- u_{{3}{6}} l_{{6}{3}} $$
$$ u_{{2}{3}} = \dfrac{a_{{2}{3}}- u_{{2}{4}} l_{{4}{3}}- u_{{2}{5}} l_{{5}{3}}- u_{{2}{6}} l_{{6}{3}}}{l_{{3}{3}}} $$
$$ u_{{1}{3}} = \dfrac{a_{{1}{3}}- u_{{1}{4}} l_{{4}{3}}- u_{{1}{5}} l_{{5}{3}}- u_{{1}{6}} l_{{6}{3}}}{l_{{3}{3}}} $$
$$ l_{{6}{2}} = a_{{6}{2}} $$
$$ l_{{5}{2}} = a_{{5}{2}}- u_{{5}{6}} l_{{6}{2}} $$
$$ l_{{4}{2}} = a_{{4}{2}}- u_{{4}{5}} l_{{5}{2}}- u_{{4}{6}} l_{{6}{2}} $$
$$ l_{{3}{2}} = a_{{3}{2}}- u_{{3}{4}} l_{{4}{2}}- u_{{3}{5}} l_{{5}{2}}- u_{{3}{6}} l_{{6}{2}} $$
$$ l_{{2}{2}} = a_{{2}{2}}- u_{{2}{3}} l_{{3}{2}}- u_{{2}{4}} l_{{4}{2}}- u_{{2}{5}} l_{{5}{2}}- u_{{2}{6}} l_{{6}{2}} $$
$$ u_{{1}{2}} = \dfrac{a_{{1}{2}}- u_{{1}{3}} l_{{3}{2}}- u_{{1}{4}} l_{{4}{2}}- u_{{1}{5}} l_{{5}{2}}- u_{{1}{6}} l_{{6}{2}}}{l_{{2}{2}}} $$
$$ l_{{6}{1}} = a_{{6}{1}} $$
$$ l_{{5}{1}} = a_{{5}{1}}- u_{{5}{6}} l_{{6}{1}} $$
$$ l_{{4}{1}} = a_{{4}{1}}- u_{{4}{5}} l_{{5}{1}}- u_{{4}{6}} l_{{6}{1}} $$
$$ l_{{3}{1}} = a_{{3}{1}}- u_{{3}{4}} l_{{4}{1}}- u_{{3}{5}} l_{{5}{1}}- u_{{3}{6}} l_{{6}{1}} $$
$$ l_{{2}{1}} = a_{{2}{1}}- u_{{2}{3}} l_{{3}{1}}- u_{{2}{4}} l_{{4}{1}}- u_{{2}{5}} l_{{5}{1}}- u_{{2}{6}} l_{{6}{1}} $$
$$ l_{{1}{1}} = a_{{1}{1}}- u_{{1}{2}} l_{{2}{1}}- u_{{1}{3}} l_{{3}{1}}- u_{{1}{4}} l_{{4}{1}}- u_{{1}{5}} l_{{5}{1}}- u_{{1}{6}} l_{{6}{1}} $$
\end{multicols}
\section{Metodo Ul}
\subsection{Caso de matrices de $2\times 2$ }
$$ \left( 
\begin{array}{cc}
a_{{1}{1}} & a_{{1}{2}} \\
a_{{2}{1}} & a_{{2}{2}} 
 \end{array}
\right)
 = \left( 
\begin{array}{cc}
u_{{1}{1}} & u_{{1}{2}} \\
0 & u_{{2}{2}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{cc}
1 & 0 \\
l_{{2}{1}} & 1 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{cc}
u_{{1}{1}} & u_{{1}{2}} \\
l_{{2}{1}} & u_{{2}{2}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{2}{2}} = u_{{2}{2}} $$
$$ a_{{1}{2}} = u_{{1}{2}} $$
$$ a_{{2}{1}} = u_{{2}{2}} l_{{2}{1}} $$
$$ a_{{1}{1}} = u_{{1}{1}} + u_{{1}{2}} l_{{2}{1}} $$
\vfill\null
\columnbreak
$$ u_{{2}{2}} = a_{{2}{2}} $$
$$ u_{{1}{2}} = a_{{1}{2}} $$
$$ l_{{2}{1}} = \dfrac{a_{{2}{1}}}{u_{{2}{2}}} $$
$$ u_{{1}{1}} = a_{{1}{1}}- u_{{1}{2}} l_{{2}{1}} $$
\end{multicols}
\subsection{Caso de matrices de $3\times 3$ }
$$ \left( 
\begin{array}{ccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} \\
0 & u_{{2}{2}} & u_{{2}{3}} \\
0 & 0 & u_{{3}{3}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{ccc}
1 & 0 & 0 \\
l_{{2}{1}} & 1 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & 1 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{ccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} \\
l_{{2}{1}} & u_{{2}{2}} & u_{{2}{3}} \\
l_{{3}{1}} & l_{{3}{2}} & u_{{3}{3}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{3}{3}} = u_{{3}{3}} $$
$$ a_{{2}{3}} = u_{{2}{3}} $$
$$ a_{{1}{3}} = u_{{1}{3}} $$
$$ a_{{3}{2}} = u_{{3}{3}} l_{{3}{2}} $$
$$ a_{{2}{2}} = u_{{2}{2}} + u_{{2}{3}} l_{{3}{2}} $$
$$ a_{{1}{2}} = u_{{1}{2}} + u_{{1}{3}} l_{{3}{2}} $$
$$ a_{{3}{1}} = u_{{3}{3}} l_{{3}{1}} $$
$$ a_{{2}{1}} = u_{{2}{2}} l_{{2}{1}} + u_{{2}{3}} l_{{3}{1}} $$
$$ a_{{1}{1}} = u_{{1}{1}} + u_{{1}{2}} l_{{2}{1}} + u_{{1}{3}} l_{{3}{1}} $$
\vfill\null
\columnbreak
$$ u_{{3}{3}} = a_{{3}{3}} $$
$$ u_{{2}{3}} = a_{{2}{3}} $$
$$ u_{{1}{3}} = a_{{1}{3}} $$
$$ l_{{3}{2}} = \dfrac{a_{{3}{2}}}{u_{{3}{3}}} $$
$$ u_{{2}{2}} = a_{{2}{2}}- u_{{2}{3}} l_{{3}{2}} $$
$$ u_{{1}{2}} = a_{{1}{2}}- u_{{1}{3}} l_{{3}{2}} $$
$$ l_{{3}{1}} = \dfrac{a_{{3}{1}}}{u_{{3}{3}}} $$
$$ l_{{2}{1}} = \dfrac{a_{{2}{1}}- u_{{2}{3}} l_{{3}{1}}}{u_{{2}{2}}} $$
$$ u_{{1}{1}} = a_{{1}{1}}- u_{{1}{2}} l_{{2}{1}}- u_{{1}{3}} l_{{3}{1}} $$
\end{multicols}
\subsection{Caso de matrices de $4\times 4$ }
$$ \left( 
\begin{array}{cccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} & a_{{1}{4}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} & a_{{2}{4}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} & a_{{3}{4}} \\
a_{{4}{1}} & a_{{4}{2}} & a_{{4}{3}} & a_{{4}{4}} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} \\
0 & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} \\
0 & 0 & u_{{3}{3}} & u_{{3}{4}} \\
0 & 0 & 0 & u_{{4}{4}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
l_{{2}{1}} & 1 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & 1 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & 1 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{cccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} \\
l_{{2}{1}} & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} \\
l_{{3}{1}} & l_{{3}{2}} & u_{{3}{3}} & u_{{3}{4}} \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & u_{{4}{4}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{4}{4}} = u_{{4}{4}} $$
$$ a_{{3}{4}} = u_{{3}{4}} $$
$$ a_{{2}{4}} = u_{{2}{4}} $$
$$ a_{{1}{4}} = u_{{1}{4}} $$
$$ a_{{4}{3}} = u_{{4}{4}} l_{{4}{3}} $$
$$ a_{{3}{3}} = u_{{3}{3}} + u_{{3}{4}} l_{{4}{3}} $$
$$ a_{{2}{3}} = u_{{2}{3}} + u_{{2}{4}} l_{{4}{3}} $$
$$ a_{{1}{3}} = u_{{1}{3}} + u_{{1}{4}} l_{{4}{3}} $$
$$ a_{{4}{2}} = u_{{4}{4}} l_{{4}{2}} $$
$$ a_{{3}{2}} = u_{{3}{3}} l_{{3}{2}} + u_{{3}{4}} l_{{4}{2}} $$
$$ a_{{2}{2}} = u_{{2}{2}} + u_{{2}{3}} l_{{3}{2}} + u_{{2}{4}} l_{{4}{2}} $$
$$ a_{{1}{2}} = u_{{1}{2}} + u_{{1}{3}} l_{{3}{2}} + u_{{1}{4}} l_{{4}{2}} $$
$$ a_{{4}{1}} = u_{{4}{4}} l_{{4}{1}} $$
$$ a_{{3}{1}} = u_{{3}{3}} l_{{3}{1}} + u_{{3}{4}} l_{{4}{1}} $$
$$ a_{{2}{1}} = u_{{2}{2}} l_{{2}{1}} + u_{{2}{3}} l_{{3}{1}} + u_{{2}{4}} l_{{4}{1}} $$
$$ a_{{1}{1}} = u_{{1}{1}} + u_{{1}{2}} l_{{2}{1}} + u_{{1}{3}} l_{{3}{1}} + u_{{1}{4}} l_{{4}{1}} $$
\vfill\null
\columnbreak
$$ u_{{4}{4}} = a_{{4}{4}} $$
$$ u_{{3}{4}} = a_{{3}{4}} $$
$$ u_{{2}{4}} = a_{{2}{4}} $$
$$ u_{{1}{4}} = a_{{1}{4}} $$
$$ l_{{4}{3}} = \dfrac{a_{{4}{3}}}{u_{{4}{4}}} $$
$$ u_{{3}{3}} = a_{{3}{3}}- u_{{3}{4}} l_{{4}{3}} $$
$$ u_{{2}{3}} = a_{{2}{3}}- u_{{2}{4}} l_{{4}{3}} $$
$$ u_{{1}{3}} = a_{{1}{3}}- u_{{1}{4}} l_{{4}{3}} $$
$$ l_{{4}{2}} = \dfrac{a_{{4}{2}}}{u_{{4}{4}}} $$
$$ l_{{3}{2}} = \dfrac{a_{{3}{2}}- u_{{3}{4}} l_{{4}{2}}}{u_{{3}{3}}} $$
$$ u_{{2}{2}} = a_{{2}{2}}- u_{{2}{3}} l_{{3}{2}}- u_{{2}{4}} l_{{4}{2}} $$
$$ u_{{1}{2}} = a_{{1}{2}}- u_{{1}{3}} l_{{3}{2}}- u_{{1}{4}} l_{{4}{2}} $$
$$ l_{{4}{1}} = \dfrac{a_{{4}{1}}}{u_{{4}{4}}} $$
$$ l_{{3}{1}} = \dfrac{a_{{3}{1}}- u_{{3}{4}} l_{{4}{1}}}{u_{{3}{3}}} $$
$$ l_{{2}{1}} = \dfrac{a_{{2}{1}}- u_{{2}{3}} l_{{3}{1}}- u_{{2}{4}} l_{{4}{1}}}{u_{{2}{2}}} $$
$$ u_{{1}{1}} = a_{{1}{1}}- u_{{1}{2}} l_{{2}{1}}- u_{{1}{3}} l_{{3}{1}}- u_{{1}{4}} l_{{4}{1}} $$
\end{multicols}
\subsection{Caso de matrices de $5\times 5$ }
$$ \left( 
\begin{array}{ccccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} & a_{{1}{4}} & a_{{1}{5}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} & a_{{2}{4}} & a_{{2}{5}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} & a_{{3}{4}} & a_{{3}{5}} \\
a_{{4}{1}} & a_{{4}{2}} & a_{{4}{3}} & a_{{4}{4}} & a_{{4}{5}} \\
a_{{5}{1}} & a_{{5}{2}} & a_{{5}{3}} & a_{{5}{4}} & a_{{5}{5}} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} \\
0 & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} \\
0 & 0 & u_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} \\
0 & 0 & 0 & u_{{4}{4}} & u_{{4}{5}} \\
0 & 0 & 0 & 0 & u_{{5}{5}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
l_{{2}{1}} & 1 & 0 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & 1 & 0 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & 1 & 0 \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & 1 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{ccccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} \\
l_{{2}{1}} & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} \\
l_{{3}{1}} & l_{{3}{2}} & u_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & u_{{4}{4}} & u_{{4}{5}} \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & u_{{5}{5}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{5}{5}} = u_{{5}{5}} $$
$$ a_{{4}{5}} = u_{{4}{5}} $$
$$ a_{{3}{5}} = u_{{3}{5}} $$
$$ a_{{2}{5}} = u_{{2}{5}} $$
$$ a_{{1}{5}} = u_{{1}{5}} $$
$$ a_{{5}{4}} = u_{{5}{5}} l_{{5}{4}} $$
$$ a_{{4}{4}} = u_{{4}{4}} + u_{{4}{5}} l_{{5}{4}} $$
$$ a_{{3}{4}} = u_{{3}{4}} + u_{{3}{5}} l_{{5}{4}} $$
$$ a_{{2}{4}} = u_{{2}{4}} + u_{{2}{5}} l_{{5}{4}} $$
$$ a_{{1}{4}} = u_{{1}{4}} + u_{{1}{5}} l_{{5}{4}} $$
$$ a_{{5}{3}} = u_{{5}{5}} l_{{5}{3}} $$
$$ a_{{4}{3}} = u_{{4}{4}} l_{{4}{3}} + u_{{4}{5}} l_{{5}{3}} $$
$$ a_{{3}{3}} = u_{{3}{3}} + u_{{3}{4}} l_{{4}{3}} + u_{{3}{5}} l_{{5}{3}} $$
$$ a_{{2}{3}} = u_{{2}{3}} + u_{{2}{4}} l_{{4}{3}} + u_{{2}{5}} l_{{5}{3}} $$
$$ a_{{1}{3}} = u_{{1}{3}} + u_{{1}{4}} l_{{4}{3}} + u_{{1}{5}} l_{{5}{3}} $$
$$ a_{{5}{2}} = u_{{5}{5}} l_{{5}{2}} $$
$$ a_{{4}{2}} = u_{{4}{4}} l_{{4}{2}} + u_{{4}{5}} l_{{5}{2}} $$
$$ a_{{3}{2}} = u_{{3}{3}} l_{{3}{2}} + u_{{3}{4}} l_{{4}{2}} + u_{{3}{5}} l_{{5}{2}} $$
$$ a_{{2}{2}} = u_{{2}{2}} + u_{{2}{3}} l_{{3}{2}} + u_{{2}{4}} l_{{4}{2}} + u_{{2}{5}} l_{{5}{2}} $$
$$ a_{{1}{2}} = u_{{1}{2}} + u_{{1}{3}} l_{{3}{2}} + u_{{1}{4}} l_{{4}{2}} + u_{{1}{5}} l_{{5}{2}} $$
$$ a_{{5}{1}} = u_{{5}{5}} l_{{5}{1}} $$
$$ a_{{4}{1}} = u_{{4}{4}} l_{{4}{1}} + u_{{4}{5}} l_{{5}{1}} $$
$$ a_{{3}{1}} = u_{{3}{3}} l_{{3}{1}} + u_{{3}{4}} l_{{4}{1}} + u_{{3}{5}} l_{{5}{1}} $$
$$ a_{{2}{1}} = u_{{2}{2}} l_{{2}{1}} + u_{{2}{3}} l_{{3}{1}} + u_{{2}{4}} l_{{4}{1}} + u_{{2}{5}} l_{{5}{1}} $$
$$ a_{{1}{1}} = u_{{1}{1}} + u_{{1}{2}} l_{{2}{1}} + u_{{1}{3}} l_{{3}{1}} + u_{{1}{4}} l_{{4}{1}} + u_{{1}{5}} l_{{5}{1}} $$
\vfill\null
\columnbreak
$$ u_{{5}{5}} = a_{{5}{5}} $$
$$ u_{{4}{5}} = a_{{4}{5}} $$
$$ u_{{3}{5}} = a_{{3}{5}} $$
$$ u_{{2}{5}} = a_{{2}{5}} $$
$$ u_{{1}{5}} = a_{{1}{5}} $$
$$ l_{{5}{4}} = \dfrac{a_{{5}{4}}}{u_{{5}{5}}} $$
$$ u_{{4}{4}} = a_{{4}{4}}- u_{{4}{5}} l_{{5}{4}} $$
$$ u_{{3}{4}} = a_{{3}{4}}- u_{{3}{5}} l_{{5}{4}} $$
$$ u_{{2}{4}} = a_{{2}{4}}- u_{{2}{5}} l_{{5}{4}} $$
$$ u_{{1}{4}} = a_{{1}{4}}- u_{{1}{5}} l_{{5}{4}} $$
$$ l_{{5}{3}} = \dfrac{a_{{5}{3}}}{u_{{5}{5}}} $$
$$ l_{{4}{3}} = \dfrac{a_{{4}{3}}- u_{{4}{5}} l_{{5}{3}}}{u_{{4}{4}}} $$
$$ u_{{3}{3}} = a_{{3}{3}}- u_{{3}{4}} l_{{4}{3}}- u_{{3}{5}} l_{{5}{3}} $$
$$ u_{{2}{3}} = a_{{2}{3}}- u_{{2}{4}} l_{{4}{3}}- u_{{2}{5}} l_{{5}{3}} $$
$$ u_{{1}{3}} = a_{{1}{3}}- u_{{1}{4}} l_{{4}{3}}- u_{{1}{5}} l_{{5}{3}} $$
$$ l_{{5}{2}} = \dfrac{a_{{5}{2}}}{u_{{5}{5}}} $$
$$ l_{{4}{2}} = \dfrac{a_{{4}{2}}- u_{{4}{5}} l_{{5}{2}}}{u_{{4}{4}}} $$
$$ l_{{3}{2}} = \dfrac{a_{{3}{2}}- u_{{3}{4}} l_{{4}{2}}- u_{{3}{5}} l_{{5}{2}}}{u_{{3}{3}}} $$
$$ u_{{2}{2}} = a_{{2}{2}}- u_{{2}{3}} l_{{3}{2}}- u_{{2}{4}} l_{{4}{2}}- u_{{2}{5}} l_{{5}{2}} $$
$$ u_{{1}{2}} = a_{{1}{2}}- u_{{1}{3}} l_{{3}{2}}- u_{{1}{4}} l_{{4}{2}}- u_{{1}{5}} l_{{5}{2}} $$
$$ l_{{5}{1}} = \dfrac{a_{{5}{1}}}{u_{{5}{5}}} $$
$$ l_{{4}{1}} = \dfrac{a_{{4}{1}}- u_{{4}{5}} l_{{5}{1}}}{u_{{4}{4}}} $$
$$ l_{{3}{1}} = \dfrac{a_{{3}{1}}- u_{{3}{4}} l_{{4}{1}}- u_{{3}{5}} l_{{5}{1}}}{u_{{3}{3}}} $$
$$ l_{{2}{1}} = \dfrac{a_{{2}{1}}- u_{{2}{3}} l_{{3}{1}}- u_{{2}{4}} l_{{4}{1}}- u_{{2}{5}} l_{{5}{1}}}{u_{{2}{2}}} $$
$$ u_{{1}{1}} = a_{{1}{1}}- u_{{1}{2}} l_{{2}{1}}- u_{{1}{3}} l_{{3}{1}}- u_{{1}{4}} l_{{4}{1}}- u_{{1}{5}} l_{{5}{1}} $$
\end{multicols}
\subsection{Caso de matrices de $6\times 6$ }
$$ \left( 
\begin{array}{cccccc}
a_{{1}{1}} & a_{{1}{2}} & a_{{1}{3}} & a_{{1}{4}} & a_{{1}{5}} & a_{{1}{6}} \\
a_{{2}{1}} & a_{{2}{2}} & a_{{2}{3}} & a_{{2}{4}} & a_{{2}{5}} & a_{{2}{6}} \\
a_{{3}{1}} & a_{{3}{2}} & a_{{3}{3}} & a_{{3}{4}} & a_{{3}{5}} & a_{{3}{6}} \\
a_{{4}{1}} & a_{{4}{2}} & a_{{4}{3}} & a_{{4}{4}} & a_{{4}{5}} & a_{{4}{6}} \\
a_{{5}{1}} & a_{{5}{2}} & a_{{5}{3}} & a_{{5}{4}} & a_{{5}{5}} & a_{{5}{6}} \\
a_{{6}{1}} & a_{{6}{2}} & a_{{6}{3}} & a_{{6}{4}} & a_{{6}{5}} & a_{{6}{6}} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} & u_{{1}{6}} \\
0 & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} & u_{{2}{6}} \\
0 & 0 & u_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} & u_{{3}{6}} \\
0 & 0 & 0 & u_{{4}{4}} & u_{{4}{5}} & u_{{4}{6}} \\
0 & 0 & 0 & 0 & u_{{5}{5}} & u_{{5}{6}} \\
0 & 0 & 0 & 0 & 0 & u_{{6}{6}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0 \\
l_{{2}{1}} & 1 & 0 & 0 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & 1 & 0 & 0 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & 1 & 0 & 0 \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & 1 & 0 \\
l_{{6}{1}} & l_{{6}{2}} & l_{{6}{3}} & l_{{6}{4}} & l_{{6}{5}} & 1 
 \end{array}
\right)
 $$
$$ \left( 
\begin{array}{cccccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} & u_{{1}{6}} \\
l_{{2}{1}} & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} & u_{{2}{6}} \\
l_{{3}{1}} & l_{{3}{2}} & u_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} & u_{{3}{6}} \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & u_{{4}{4}} & u_{{4}{5}} & u_{{4}{6}} \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & u_{{5}{5}} & u_{{5}{6}} \\
l_{{6}{1}} & l_{{6}{2}} & l_{{6}{3}} & l_{{6}{4}} & l_{{6}{5}} & u_{{6}{6}} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ a_{{6}{6}} = u_{{6}{6}} $$
$$ a_{{5}{6}} = u_{{5}{6}} $$
$$ a_{{4}{6}} = u_{{4}{6}} $$
$$ a_{{3}{6}} = u_{{3}{6}} $$
$$ a_{{2}{6}} = u_{{2}{6}} $$
$$ a_{{1}{6}} = u_{{1}{6}} $$
$$ a_{{6}{5}} = u_{{6}{6}} l_{{6}{5}} $$
$$ a_{{5}{5}} = u_{{5}{5}} + u_{{5}{6}} l_{{6}{5}} $$
$$ a_{{4}{5}} = u_{{4}{5}} + u_{{4}{6}} l_{{6}{5}} $$
$$ a_{{3}{5}} = u_{{3}{5}} + u_{{3}{6}} l_{{6}{5}} $$
$$ a_{{2}{5}} = u_{{2}{5}} + u_{{2}{6}} l_{{6}{5}} $$
$$ a_{{1}{5}} = u_{{1}{5}} + u_{{1}{6}} l_{{6}{5}} $$
$$ a_{{6}{4}} = u_{{6}{6}} l_{{6}{4}} $$
$$ a_{{5}{4}} = u_{{5}{5}} l_{{5}{4}} + u_{{5}{6}} l_{{6}{4}} $$
$$ a_{{4}{4}} = u_{{4}{4}} + u_{{4}{5}} l_{{5}{4}} + u_{{4}{6}} l_{{6}{4}} $$
$$ a_{{3}{4}} = u_{{3}{4}} + u_{{3}{5}} l_{{5}{4}} + u_{{3}{6}} l_{{6}{4}} $$
$$ a_{{2}{4}} = u_{{2}{4}} + u_{{2}{5}} l_{{5}{4}} + u_{{2}{6}} l_{{6}{4}} $$
$$ a_{{1}{4}} = u_{{1}{4}} + u_{{1}{5}} l_{{5}{4}} + u_{{1}{6}} l_{{6}{4}} $$
$$ a_{{6}{3}} = u_{{6}{6}} l_{{6}{3}} $$
$$ a_{{5}{3}} = u_{{5}{5}} l_{{5}{3}} + u_{{5}{6}} l_{{6}{3}} $$
$$ a_{{4}{3}} = u_{{4}{4}} l_{{4}{3}} + u_{{4}{5}} l_{{5}{3}} + u_{{4}{6}} l_{{6}{3}} $$
$$ a_{{3}{3}} = u_{{3}{3}} + u_{{3}{4}} l_{{4}{3}} + u_{{3}{5}} l_{{5}{3}} + u_{{3}{6}} l_{{6}{3}} $$
$$ a_{{2}{3}} = u_{{2}{3}} + u_{{2}{4}} l_{{4}{3}} + u_{{2}{5}} l_{{5}{3}} + u_{{2}{6}} l_{{6}{3}} $$
$$ a_{{1}{3}} = u_{{1}{3}} + u_{{1}{4}} l_{{4}{3}} + u_{{1}{5}} l_{{5}{3}} + u_{{1}{6}} l_{{6}{3}} $$
$$ a_{{6}{2}} = u_{{6}{6}} l_{{6}{2}} $$
$$ a_{{5}{2}} = u_{{5}{5}} l_{{5}{2}} + u_{{5}{6}} l_{{6}{2}} $$
$$ a_{{4}{2}} = u_{{4}{4}} l_{{4}{2}} + u_{{4}{5}} l_{{5}{2}} + u_{{4}{6}} l_{{6}{2}} $$
$$ a_{{3}{2}} = u_{{3}{3}} l_{{3}{2}} + u_{{3}{4}} l_{{4}{2}} + u_{{3}{5}} l_{{5}{2}} + u_{{3}{6}} l_{{6}{2}} $$
$$ a_{{2}{2}} = u_{{2}{2}} + u_{{2}{3}} l_{{3}{2}} + u_{{2}{4}} l_{{4}{2}} + u_{{2}{5}} l_{{5}{2}} + u_{{2}{6}} l_{{6}{2}} $$
$$ a_{{1}{2}} = u_{{1}{2}} + u_{{1}{3}} l_{{3}{2}} + u_{{1}{4}} l_{{4}{2}} + u_{{1}{5}} l_{{5}{2}} + u_{{1}{6}} l_{{6}{2}} $$
$$ a_{{6}{1}} = u_{{6}{6}} l_{{6}{1}} $$
$$ a_{{5}{1}} = u_{{5}{5}} l_{{5}{1}} + u_{{5}{6}} l_{{6}{1}} $$
$$ a_{{4}{1}} = u_{{4}{4}} l_{{4}{1}} + u_{{4}{5}} l_{{5}{1}} + u_{{4}{6}} l_{{6}{1}} $$
$$ a_{{3}{1}} = u_{{3}{3}} l_{{3}{1}} + u_{{3}{4}} l_{{4}{1}} + u_{{3}{5}} l_{{5}{1}} + u_{{3}{6}} l_{{6}{1}} $$
$$ a_{{2}{1}} = u_{{2}{2}} l_{{2}{1}} + u_{{2}{3}} l_{{3}{1}} + u_{{2}{4}} l_{{4}{1}} + u_{{2}{5}} l_{{5}{1}} + u_{{2}{6}} l_{{6}{1}} $$
$$ a_{{1}{1}} = u_{{1}{1}} + u_{{1}{2}} l_{{2}{1}} + u_{{1}{3}} l_{{3}{1}} + u_{{1}{4}} l_{{4}{1}} + u_{{1}{5}} l_{{5}{1}} + u_{{1}{6}} l_{{6}{1}} $$
\vfill\null
\columnbreak
$$ u_{{6}{6}} = a_{{6}{6}} $$
$$ u_{{5}{6}} = a_{{5}{6}} $$
$$ u_{{4}{6}} = a_{{4}{6}} $$
$$ u_{{3}{6}} = a_{{3}{6}} $$
$$ u_{{2}{6}} = a_{{2}{6}} $$
$$ u_{{1}{6}} = a_{{1}{6}} $$
$$ l_{{6}{5}} = \dfrac{a_{{6}{5}}}{u_{{6}{6}}} $$
$$ u_{{5}{5}} = a_{{5}{5}}- u_{{5}{6}} l_{{6}{5}} $$
$$ u_{{4}{5}} = a_{{4}{5}}- u_{{4}{6}} l_{{6}{5}} $$
$$ u_{{3}{5}} = a_{{3}{5}}- u_{{3}{6}} l_{{6}{5}} $$
$$ u_{{2}{5}} = a_{{2}{5}}- u_{{2}{6}} l_{{6}{5}} $$
$$ u_{{1}{5}} = a_{{1}{5}}- u_{{1}{6}} l_{{6}{5}} $$
$$ l_{{6}{4}} = \dfrac{a_{{6}{4}}}{u_{{6}{6}}} $$
$$ l_{{5}{4}} = \dfrac{a_{{5}{4}}- u_{{5}{6}} l_{{6}{4}}}{u_{{5}{5}}} $$
$$ u_{{4}{4}} = a_{{4}{4}}- u_{{4}{5}} l_{{5}{4}}- u_{{4}{6}} l_{{6}{4}} $$
$$ u_{{3}{4}} = a_{{3}{4}}- u_{{3}{5}} l_{{5}{4}}- u_{{3}{6}} l_{{6}{4}} $$
$$ u_{{2}{4}} = a_{{2}{4}}- u_{{2}{5}} l_{{5}{4}}- u_{{2}{6}} l_{{6}{4}} $$
$$ u_{{1}{4}} = a_{{1}{4}}- u_{{1}{5}} l_{{5}{4}}- u_{{1}{6}} l_{{6}{4}} $$
$$ l_{{6}{3}} = \dfrac{a_{{6}{3}}}{u_{{6}{6}}} $$
$$ l_{{5}{3}} = \dfrac{a_{{5}{3}}- u_{{5}{6}} l_{{6}{3}}}{u_{{5}{5}}} $$
$$ l_{{4}{3}} = \dfrac{a_{{4}{3}}- u_{{4}{5}} l_{{5}{3}}- u_{{4}{6}} l_{{6}{3}}}{u_{{4}{4}}} $$
$$ u_{{3}{3}} = a_{{3}{3}}- u_{{3}{4}} l_{{4}{3}}- u_{{3}{5}} l_{{5}{3}}- u_{{3}{6}} l_{{6}{3}} $$
$$ u_{{2}{3}} = a_{{2}{3}}- u_{{2}{4}} l_{{4}{3}}- u_{{2}{5}} l_{{5}{3}}- u_{{2}{6}} l_{{6}{3}} $$
$$ u_{{1}{3}} = a_{{1}{3}}- u_{{1}{4}} l_{{4}{3}}- u_{{1}{5}} l_{{5}{3}}- u_{{1}{6}} l_{{6}{3}} $$
$$ l_{{6}{2}} = \dfrac{a_{{6}{2}}}{u_{{6}{6}}} $$
$$ l_{{5}{2}} = \dfrac{a_{{5}{2}}- u_{{5}{6}} l_{{6}{2}}}{u_{{5}{5}}} $$
$$ l_{{4}{2}} = \dfrac{a_{{4}{2}}- u_{{4}{5}} l_{{5}{2}}- u_{{4}{6}} l_{{6}{2}}}{u_{{4}{4}}} $$
$$ l_{{3}{2}} = \dfrac{a_{{3}{2}}- u_{{3}{4}} l_{{4}{2}}- u_{{3}{5}} l_{{5}{2}}- u_{{3}{6}} l_{{6}{2}}}{u_{{3}{3}}} $$
$$ u_{{2}{2}} = a_{{2}{2}}- u_{{2}{3}} l_{{3}{2}}- u_{{2}{4}} l_{{4}{2}}- u_{{2}{5}} l_{{5}{2}}- u_{{2}{6}} l_{{6}{2}} $$
$$ u_{{1}{2}} = a_{{1}{2}}- u_{{1}{3}} l_{{3}{2}}- u_{{1}{4}} l_{{4}{2}}- u_{{1}{5}} l_{{5}{2}}- u_{{1}{6}} l_{{6}{2}} $$
$$ l_{{6}{1}} = \dfrac{a_{{6}{1}}}{u_{{6}{6}}} $$
$$ l_{{5}{1}} = \dfrac{a_{{5}{1}}- u_{{5}{6}} l_{{6}{1}}}{u_{{5}{5}}} $$
$$ l_{{4}{1}} = \dfrac{a_{{4}{1}}- u_{{4}{5}} l_{{5}{1}}- u_{{4}{6}} l_{{6}{1}}}{u_{{4}{4}}} $$
$$ l_{{3}{1}} = \dfrac{a_{{3}{1}}- u_{{3}{4}} l_{{4}{1}}- u_{{3}{5}} l_{{5}{1}}- u_{{3}{6}} l_{{6}{1}}}{u_{{3}{3}}} $$
$$ l_{{2}{1}} = \dfrac{a_{{2}{1}}- u_{{2}{3}} l_{{3}{1}}- u_{{2}{4}} l_{{4}{1}}- u_{{2}{5}} l_{{5}{1}}- u_{{2}{6}} l_{{6}{1}}}{u_{{2}{2}}} $$
$$ u_{{1}{1}} = a_{{1}{1}}- u_{{1}{2}} l_{{2}{1}}- u_{{1}{3}} l_{{3}{1}}- u_{{1}{4}} l_{{4}{1}}- u_{{1}{5}} l_{{5}{1}}- u_{{1}{6}} l_{{6}{1}} $$
\end{multicols}
\section{Resoluci\'on de matrices triangulares L}
\subsection{Caso de matrices de $2\times 2$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} 
 \end{array}
\right)
 = \left( 
\begin{array}{cc}
l_{{1}{1}} & 0 \\
l_{{2}{1}} & l_{{2}{2}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{1} = l_{{1}{1}} x_{1} $$
$$ y_{2} = l_{{2}{1}} x_{1} + l_{{2}{2}} x_{2} $$
\vfill\null
\columnbreak
$$ x_{1} = \dfrac{y_{1}}{l_{{1}{1}}} $$
$$ x_{2} = \dfrac{y_{2}- l_{{2}{1}} x_{1}}{l_{{2}{2}}} $$
\end{multicols}
\subsection{Caso de matrices de $3\times 3$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccc}
l_{{1}{1}} & 0 & 0 \\
l_{{2}{1}} & l_{{2}{2}} & 0 \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{1} = l_{{1}{1}} x_{1} $$
$$ y_{2} = l_{{2}{1}} x_{1} + l_{{2}{2}} x_{2} $$
$$ y_{3} = l_{{3}{1}} x_{1} + l_{{3}{2}} x_{2} + l_{{3}{3}} x_{3} $$
\vfill\null
\columnbreak
$$ x_{1} = \dfrac{y_{1}}{l_{{1}{1}}} $$
$$ x_{2} = \dfrac{y_{2}- l_{{2}{1}} x_{1}}{l_{{2}{2}}} $$
$$ x_{3} = \dfrac{y_{3}- l_{{3}{1}} x_{1}- l_{{3}{2}} x_{2}}{l_{{3}{3}}} $$
\end{multicols}
\subsection{Caso de matrices de $4\times 4$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} \\
y_{4} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccc}
l_{{1}{1}} & 0 & 0 & 0 \\
l_{{2}{1}} & l_{{2}{2}} & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{1} = l_{{1}{1}} x_{1} $$
$$ y_{2} = l_{{2}{1}} x_{1} + l_{{2}{2}} x_{2} $$
$$ y_{3} = l_{{3}{1}} x_{1} + l_{{3}{2}} x_{2} + l_{{3}{3}} x_{3} $$
$$ y_{4} = l_{{4}{1}} x_{1} + l_{{4}{2}} x_{2} + l_{{4}{3}} x_{3} + l_{{4}{4}} x_{4} $$
\vfill\null
\columnbreak
$$ x_{1} = \dfrac{y_{1}}{l_{{1}{1}}} $$
$$ x_{2} = \dfrac{y_{2}- l_{{2}{1}} x_{1}}{l_{{2}{2}}} $$
$$ x_{3} = \dfrac{y_{3}- l_{{3}{1}} x_{1}- l_{{3}{2}} x_{2}}{l_{{3}{3}}} $$
$$ x_{4} = \dfrac{y_{4}- l_{{4}{1}} x_{1}- l_{{4}{2}} x_{2}- l_{{4}{3}} x_{3}}{l_{{4}{4}}} $$
\end{multicols}
\subsection{Caso de matrices de $5\times 5$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} \\
y_{4} \\
y_{5} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccccc}
l_{{1}{1}} & 0 & 0 & 0 & 0 \\
l_{{2}{1}} & l_{{2}{2}} & 0 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & 0 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} & 0 \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & l_{{5}{5}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} \\
x_{5} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{1} = l_{{1}{1}} x_{1} $$
$$ y_{2} = l_{{2}{1}} x_{1} + l_{{2}{2}} x_{2} $$
$$ y_{3} = l_{{3}{1}} x_{1} + l_{{3}{2}} x_{2} + l_{{3}{3}} x_{3} $$
$$ y_{4} = l_{{4}{1}} x_{1} + l_{{4}{2}} x_{2} + l_{{4}{3}} x_{3} + l_{{4}{4}} x_{4} $$
$$ y_{5} = l_{{5}{1}} x_{1} + l_{{5}{2}} x_{2} + l_{{5}{3}} x_{3} + l_{{5}{4}} x_{4} + l_{{5}{5}} x_{5} $$
\vfill\null
\columnbreak
$$ x_{1} = \dfrac{y_{1}}{l_{{1}{1}}} $$
$$ x_{2} = \dfrac{y_{2}- l_{{2}{1}} x_{1}}{l_{{2}{2}}} $$
$$ x_{3} = \dfrac{y_{3}- l_{{3}{1}} x_{1}- l_{{3}{2}} x_{2}}{l_{{3}{3}}} $$
$$ x_{4} = \dfrac{y_{4}- l_{{4}{1}} x_{1}- l_{{4}{2}} x_{2}- l_{{4}{3}} x_{3}}{l_{{4}{4}}} $$
$$ x_{5} = \dfrac{y_{5}- l_{{5}{1}} x_{1}- l_{{5}{2}} x_{2}- l_{{5}{3}} x_{3}- l_{{5}{4}} x_{4}}{l_{{5}{5}}} $$
\end{multicols}
\subsection{Caso de matrices de $6\times 6$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} \\
y_{4} \\
y_{5} \\
y_{6} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccccc}
l_{{1}{1}} & 0 & 0 & 0 & 0 & 0 \\
l_{{2}{1}} & l_{{2}{2}} & 0 & 0 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & l_{{3}{3}} & 0 & 0 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & l_{{4}{4}} & 0 & 0 \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & l_{{5}{5}} & 0 \\
l_{{6}{1}} & l_{{6}{2}} & l_{{6}{3}} & l_{{6}{4}} & l_{{6}{5}} & l_{{6}{6}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} \\
x_{5} \\
x_{6} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{1} = l_{{1}{1}} x_{1} $$
$$ y_{2} = l_{{2}{1}} x_{1} + l_{{2}{2}} x_{2} $$
$$ y_{3} = l_{{3}{1}} x_{1} + l_{{3}{2}} x_{2} + l_{{3}{3}} x_{3} $$
$$ y_{4} = l_{{4}{1}} x_{1} + l_{{4}{2}} x_{2} + l_{{4}{3}} x_{3} + l_{{4}{4}} x_{4} $$
$$ y_{5} = l_{{5}{1}} x_{1} + l_{{5}{2}} x_{2} + l_{{5}{3}} x_{3} + l_{{5}{4}} x_{4} + l_{{5}{5}} x_{5} $$
$$ y_{6} = l_{{6}{1}} x_{1} + l_{{6}{2}} x_{2} + l_{{6}{3}} x_{3} + l_{{6}{4}} x_{4} + l_{{6}{5}} x_{5} + l_{{6}{6}} x_{6} $$
\vfill\null
\columnbreak
$$ x_{1} = \dfrac{y_{1}}{l_{{1}{1}}} $$
$$ x_{2} = \dfrac{y_{2}- l_{{2}{1}} x_{1}}{l_{{2}{2}}} $$
$$ x_{3} = \dfrac{y_{3}- l_{{3}{1}} x_{1}- l_{{3}{2}} x_{2}}{l_{{3}{3}}} $$
$$ x_{4} = \dfrac{y_{4}- l_{{4}{1}} x_{1}- l_{{4}{2}} x_{2}- l_{{4}{3}} x_{3}}{l_{{4}{4}}} $$
$$ x_{5} = \dfrac{y_{5}- l_{{5}{1}} x_{1}- l_{{5}{2}} x_{2}- l_{{5}{3}} x_{3}- l_{{5}{4}} x_{4}}{l_{{5}{5}}} $$
$$ x_{6} = \dfrac{y_{6}- l_{{6}{1}} x_{1}- l_{{6}{2}} x_{2}- l_{{6}{3}} x_{3}- l_{{6}{4}} x_{4}- l_{{6}{5}} x_{5}}{l_{{6}{6}}} $$
\end{multicols}
\section{Resoluci\'on de matrices triangulares l}
\subsection{Caso de matrices de $2\times 2$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} 
 \end{array}
\right)
 = \left( 
\begin{array}{cc}
1 & 0 \\
l_{{2}{1}} & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{1} = x_{1} $$
$$ y_{2} = l_{{2}{1}} x_{1} + x_{2} $$
\vfill\null
\columnbreak
$$ x_{1} = y_{1} $$
$$ x_{2} = y_{2}- l_{{2}{1}} x_{1} $$
\end{multicols}
\subsection{Caso de matrices de $3\times 3$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccc}
1 & 0 & 0 \\
l_{{2}{1}} & 1 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{1} = x_{1} $$
$$ y_{2} = l_{{2}{1}} x_{1} + x_{2} $$
$$ y_{3} = l_{{3}{1}} x_{1} + l_{{3}{2}} x_{2} + x_{3} $$
\vfill\null
\columnbreak
$$ x_{1} = y_{1} $$
$$ x_{2} = y_{2}- l_{{2}{1}} x_{1} $$
$$ x_{3} = y_{3}- l_{{3}{1}} x_{1}- l_{{3}{2}} x_{2} $$
\end{multicols}
\subsection{Caso de matrices de $4\times 4$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} \\
y_{4} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
l_{{2}{1}} & 1 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & 1 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{1} = x_{1} $$
$$ y_{2} = l_{{2}{1}} x_{1} + x_{2} $$
$$ y_{3} = l_{{3}{1}} x_{1} + l_{{3}{2}} x_{2} + x_{3} $$
$$ y_{4} = l_{{4}{1}} x_{1} + l_{{4}{2}} x_{2} + l_{{4}{3}} x_{3} + x_{4} $$
\vfill\null
\columnbreak
$$ x_{1} = y_{1} $$
$$ x_{2} = y_{2}- l_{{2}{1}} x_{1} $$
$$ x_{3} = y_{3}- l_{{3}{1}} x_{1}- l_{{3}{2}} x_{2} $$
$$ x_{4} = y_{4}- l_{{4}{1}} x_{1}- l_{{4}{2}} x_{2}- l_{{4}{3}} x_{3} $$
\end{multicols}
\subsection{Caso de matrices de $5\times 5$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} \\
y_{4} \\
y_{5} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
l_{{2}{1}} & 1 & 0 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & 1 & 0 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & 1 & 0 \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} \\
x_{5} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{1} = x_{1} $$
$$ y_{2} = l_{{2}{1}} x_{1} + x_{2} $$
$$ y_{3} = l_{{3}{1}} x_{1} + l_{{3}{2}} x_{2} + x_{3} $$
$$ y_{4} = l_{{4}{1}} x_{1} + l_{{4}{2}} x_{2} + l_{{4}{3}} x_{3} + x_{4} $$
$$ y_{5} = l_{{5}{1}} x_{1} + l_{{5}{2}} x_{2} + l_{{5}{3}} x_{3} + l_{{5}{4}} x_{4} + x_{5} $$
\vfill\null
\columnbreak
$$ x_{1} = y_{1} $$
$$ x_{2} = y_{2}- l_{{2}{1}} x_{1} $$
$$ x_{3} = y_{3}- l_{{3}{1}} x_{1}- l_{{3}{2}} x_{2} $$
$$ x_{4} = y_{4}- l_{{4}{1}} x_{1}- l_{{4}{2}} x_{2}- l_{{4}{3}} x_{3} $$
$$ x_{5} = y_{5}- l_{{5}{1}} x_{1}- l_{{5}{2}} x_{2}- l_{{5}{3}} x_{3}- l_{{5}{4}} x_{4} $$
\end{multicols}
\subsection{Caso de matrices de $6\times 6$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} \\
y_{4} \\
y_{5} \\
y_{6} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0 \\
l_{{2}{1}} & 1 & 0 & 0 & 0 & 0 \\
l_{{3}{1}} & l_{{3}{2}} & 1 & 0 & 0 & 0 \\
l_{{4}{1}} & l_{{4}{2}} & l_{{4}{3}} & 1 & 0 & 0 \\
l_{{5}{1}} & l_{{5}{2}} & l_{{5}{3}} & l_{{5}{4}} & 1 & 0 \\
l_{{6}{1}} & l_{{6}{2}} & l_{{6}{3}} & l_{{6}{4}} & l_{{6}{5}} & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} \\
x_{5} \\
x_{6} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{1} = x_{1} $$
$$ y_{2} = l_{{2}{1}} x_{1} + x_{2} $$
$$ y_{3} = l_{{3}{1}} x_{1} + l_{{3}{2}} x_{2} + x_{3} $$
$$ y_{4} = l_{{4}{1}} x_{1} + l_{{4}{2}} x_{2} + l_{{4}{3}} x_{3} + x_{4} $$
$$ y_{5} = l_{{5}{1}} x_{1} + l_{{5}{2}} x_{2} + l_{{5}{3}} x_{3} + l_{{5}{4}} x_{4} + x_{5} $$
$$ y_{6} = l_{{6}{1}} x_{1} + l_{{6}{2}} x_{2} + l_{{6}{3}} x_{3} + l_{{6}{4}} x_{4} + l_{{6}{5}} x_{5} + x_{6} $$
\vfill\null
\columnbreak
$$ x_{1} = y_{1} $$
$$ x_{2} = y_{2}- l_{{2}{1}} x_{1} $$
$$ x_{3} = y_{3}- l_{{3}{1}} x_{1}- l_{{3}{2}} x_{2} $$
$$ x_{4} = y_{4}- l_{{4}{1}} x_{1}- l_{{4}{2}} x_{2}- l_{{4}{3}} x_{3} $$
$$ x_{5} = y_{5}- l_{{5}{1}} x_{1}- l_{{5}{2}} x_{2}- l_{{5}{3}} x_{3}- l_{{5}{4}} x_{4} $$
$$ x_{6} = y_{6}- l_{{6}{1}} x_{1}- l_{{6}{2}} x_{2}- l_{{6}{3}} x_{3}- l_{{6}{4}} x_{4}- l_{{6}{5}} x_{5} $$
\end{multicols}
\section{Resoluci\'on de matrices triangulares U}
\subsection{Caso de matrices de $2\times 2$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} 
 \end{array}
\right)
 = \left( 
\begin{array}{cc}
u_{{1}{1}} & u_{{1}{2}} \\
0 & u_{{2}{2}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{2} = u_{{2}{2}} x_{2} $$
$$ y_{1} = u_{{1}{1}} x_{1} + u_{{1}{2}} x_{2} $$
\vfill\null
\columnbreak
$$ x_{2} = \dfrac{y_{2}}{u_{{2}{2}}} $$
$$ x_{1} = \dfrac{y_{1}- u_{{1}{2}} x_{2}}{u_{{1}{1}}} $$
\end{multicols}
\subsection{Caso de matrices de $3\times 3$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} \\
0 & u_{{2}{2}} & u_{{2}{3}} \\
0 & 0 & u_{{3}{3}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{3} = u_{{3}{3}} x_{3} $$
$$ y_{2} = u_{{2}{2}} x_{2} + u_{{2}{3}} x_{3} $$
$$ y_{1} = u_{{1}{1}} x_{1} + u_{{1}{2}} x_{2} + u_{{1}{3}} x_{3} $$
\vfill\null
\columnbreak
$$ x_{3} = \dfrac{y_{3}}{u_{{3}{3}}} $$
$$ x_{2} = \dfrac{y_{2}- u_{{2}{3}} x_{3}}{u_{{2}{2}}} $$
$$ x_{1} = \dfrac{y_{1}- u_{{1}{2}} x_{2}- u_{{1}{3}} x_{3}}{u_{{1}{1}}} $$
\end{multicols}
\subsection{Caso de matrices de $4\times 4$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} \\
y_{4} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} \\
0 & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} \\
0 & 0 & u_{{3}{3}} & u_{{3}{4}} \\
0 & 0 & 0 & u_{{4}{4}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{4} = u_{{4}{4}} x_{4} $$
$$ y_{3} = u_{{3}{3}} x_{3} + u_{{3}{4}} x_{4} $$
$$ y_{2} = u_{{2}{2}} x_{2} + u_{{2}{3}} x_{3} + u_{{2}{4}} x_{4} $$
$$ y_{1} = u_{{1}{1}} x_{1} + u_{{1}{2}} x_{2} + u_{{1}{3}} x_{3} + u_{{1}{4}} x_{4} $$
\vfill\null
\columnbreak
$$ x_{4} = \dfrac{y_{4}}{u_{{4}{4}}} $$
$$ x_{3} = \dfrac{y_{3}- u_{{3}{4}} x_{4}}{u_{{3}{3}}} $$
$$ x_{2} = \dfrac{y_{2}- u_{{2}{3}} x_{3}- u_{{2}{4}} x_{4}}{u_{{2}{2}}} $$
$$ x_{1} = \dfrac{y_{1}- u_{{1}{2}} x_{2}- u_{{1}{3}} x_{3}- u_{{1}{4}} x_{4}}{u_{{1}{1}}} $$
\end{multicols}
\subsection{Caso de matrices de $5\times 5$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} \\
y_{4} \\
y_{5} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} \\
0 & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} \\
0 & 0 & u_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} \\
0 & 0 & 0 & u_{{4}{4}} & u_{{4}{5}} \\
0 & 0 & 0 & 0 & u_{{5}{5}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} \\
x_{5} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{5} = u_{{5}{5}} x_{5} $$
$$ y_{4} = u_{{4}{4}} x_{4} + u_{{4}{5}} x_{5} $$
$$ y_{3} = u_{{3}{3}} x_{3} + u_{{3}{4}} x_{4} + u_{{3}{5}} x_{5} $$
$$ y_{2} = u_{{2}{2}} x_{2} + u_{{2}{3}} x_{3} + u_{{2}{4}} x_{4} + u_{{2}{5}} x_{5} $$
$$ y_{1} = u_{{1}{1}} x_{1} + u_{{1}{2}} x_{2} + u_{{1}{3}} x_{3} + u_{{1}{4}} x_{4} + u_{{1}{5}} x_{5} $$
\vfill\null
\columnbreak
$$ x_{5} = \dfrac{y_{5}}{u_{{5}{5}}} $$
$$ x_{4} = \dfrac{y_{4}- u_{{4}{5}} x_{5}}{u_{{4}{4}}} $$
$$ x_{3} = \dfrac{y_{3}- u_{{3}{4}} x_{4}- u_{{3}{5}} x_{5}}{u_{{3}{3}}} $$
$$ x_{2} = \dfrac{y_{2}- u_{{2}{3}} x_{3}- u_{{2}{4}} x_{4}- u_{{2}{5}} x_{5}}{u_{{2}{2}}} $$
$$ x_{1} = \dfrac{y_{1}- u_{{1}{2}} x_{2}- u_{{1}{3}} x_{3}- u_{{1}{4}} x_{4}- u_{{1}{5}} x_{5}}{u_{{1}{1}}} $$
\end{multicols}
\subsection{Caso de matrices de $6\times 6$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} \\
y_{4} \\
y_{5} \\
y_{6} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccccc}
u_{{1}{1}} & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} & u_{{1}{6}} \\
0 & u_{{2}{2}} & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} & u_{{2}{6}} \\
0 & 0 & u_{{3}{3}} & u_{{3}{4}} & u_{{3}{5}} & u_{{3}{6}} \\
0 & 0 & 0 & u_{{4}{4}} & u_{{4}{5}} & u_{{4}{6}} \\
0 & 0 & 0 & 0 & u_{{5}{5}} & u_{{5}{6}} \\
0 & 0 & 0 & 0 & 0 & u_{{6}{6}} 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} \\
x_{5} \\
x_{6} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{6} = u_{{6}{6}} x_{6} $$
$$ y_{5} = u_{{5}{5}} x_{5} + u_{{5}{6}} x_{6} $$
$$ y_{4} = u_{{4}{4}} x_{4} + u_{{4}{5}} x_{5} + u_{{4}{6}} x_{6} $$
$$ y_{3} = u_{{3}{3}} x_{3} + u_{{3}{4}} x_{4} + u_{{3}{5}} x_{5} + u_{{3}{6}} x_{6} $$
$$ y_{2} = u_{{2}{2}} x_{2} + u_{{2}{3}} x_{3} + u_{{2}{4}} x_{4} + u_{{2}{5}} x_{5} + u_{{2}{6}} x_{6} $$
$$ y_{1} = u_{{1}{1}} x_{1} + u_{{1}{2}} x_{2} + u_{{1}{3}} x_{3} + u_{{1}{4}} x_{4} + u_{{1}{5}} x_{5} + u_{{1}{6}} x_{6} $$
\vfill\null
\columnbreak
$$ x_{6} = \dfrac{y_{6}}{u_{{6}{6}}} $$
$$ x_{5} = \dfrac{y_{5}- u_{{5}{6}} x_{6}}{u_{{5}{5}}} $$
$$ x_{4} = \dfrac{y_{4}- u_{{4}{5}} x_{5}- u_{{4}{6}} x_{6}}{u_{{4}{4}}} $$
$$ x_{3} = \dfrac{y_{3}- u_{{3}{4}} x_{4}- u_{{3}{5}} x_{5}- u_{{3}{6}} x_{6}}{u_{{3}{3}}} $$
$$ x_{2} = \dfrac{y_{2}- u_{{2}{3}} x_{3}- u_{{2}{4}} x_{4}- u_{{2}{5}} x_{5}- u_{{2}{6}} x_{6}}{u_{{2}{2}}} $$
$$ x_{1} = \dfrac{y_{1}- u_{{1}{2}} x_{2}- u_{{1}{3}} x_{3}- u_{{1}{4}} x_{4}- u_{{1}{5}} x_{5}- u_{{1}{6}} x_{6}}{u_{{1}{1}}} $$
\end{multicols}
\section{Resoluci\'on de matrices triangulares u}
\subsection{Caso de matrices de $2\times 2$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} 
 \end{array}
\right)
 = \left( 
\begin{array}{cc}
1 & u_{{1}{2}} \\
0 & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{2} = x_{2} $$
$$ y_{1} = x_{1} + u_{{1}{2}} x_{2} $$
\vfill\null
\columnbreak
$$ x_{2} = y_{2} $$
$$ x_{1} = y_{1}- u_{{1}{2}} x_{2} $$
\end{multicols}
\subsection{Caso de matrices de $3\times 3$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccc}
1 & u_{{1}{2}} & u_{{1}{3}} \\
0 & 1 & u_{{2}{3}} \\
0 & 0 & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{3} = x_{3} $$
$$ y_{2} = x_{2} + u_{{2}{3}} x_{3} $$
$$ y_{1} = x_{1} + u_{{1}{2}} x_{2} + u_{{1}{3}} x_{3} $$
\vfill\null
\columnbreak
$$ x_{3} = y_{3} $$
$$ x_{2} = y_{2}- u_{{2}{3}} x_{3} $$
$$ x_{1} = y_{1}- u_{{1}{2}} x_{2}- u_{{1}{3}} x_{3} $$
\end{multicols}
\subsection{Caso de matrices de $4\times 4$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} \\
y_{4} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccc}
1 & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} \\
0 & 1 & u_{{2}{3}} & u_{{2}{4}} \\
0 & 0 & 1 & u_{{3}{4}} \\
0 & 0 & 0 & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{4} = x_{4} $$
$$ y_{3} = x_{3} + u_{{3}{4}} x_{4} $$
$$ y_{2} = x_{2} + u_{{2}{3}} x_{3} + u_{{2}{4}} x_{4} $$
$$ y_{1} = x_{1} + u_{{1}{2}} x_{2} + u_{{1}{3}} x_{3} + u_{{1}{4}} x_{4} $$
\vfill\null
\columnbreak
$$ x_{4} = y_{4} $$
$$ x_{3} = y_{3}- u_{{3}{4}} x_{4} $$
$$ x_{2} = y_{2}- u_{{2}{3}} x_{3}- u_{{2}{4}} x_{4} $$
$$ x_{1} = y_{1}- u_{{1}{2}} x_{2}- u_{{1}{3}} x_{3}- u_{{1}{4}} x_{4} $$
\end{multicols}
\subsection{Caso de matrices de $5\times 5$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} \\
y_{4} \\
y_{5} 
 \end{array}
\right)
 = \left( 
\begin{array}{ccccc}
1 & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} \\
0 & 1 & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} \\
0 & 0 & 1 & u_{{3}{4}} & u_{{3}{5}} \\
0 & 0 & 0 & 1 & u_{{4}{5}} \\
0 & 0 & 0 & 0 & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} \\
x_{5} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{5} = x_{5} $$
$$ y_{4} = x_{4} + u_{{4}{5}} x_{5} $$
$$ y_{3} = x_{3} + u_{{3}{4}} x_{4} + u_{{3}{5}} x_{5} $$
$$ y_{2} = x_{2} + u_{{2}{3}} x_{3} + u_{{2}{4}} x_{4} + u_{{2}{5}} x_{5} $$
$$ y_{1} = x_{1} + u_{{1}{2}} x_{2} + u_{{1}{3}} x_{3} + u_{{1}{4}} x_{4} + u_{{1}{5}} x_{5} $$
\vfill\null
\columnbreak
$$ x_{5} = y_{5} $$
$$ x_{4} = y_{4}- u_{{4}{5}} x_{5} $$
$$ x_{3} = y_{3}- u_{{3}{4}} x_{4}- u_{{3}{5}} x_{5} $$
$$ x_{2} = y_{2}- u_{{2}{3}} x_{3}- u_{{2}{4}} x_{4}- u_{{2}{5}} x_{5} $$
$$ x_{1} = y_{1}- u_{{1}{2}} x_{2}- u_{{1}{3}} x_{3}- u_{{1}{4}} x_{4}- u_{{1}{5}} x_{5} $$
\end{multicols}
\subsection{Caso de matrices de $6\times 6$ }
$$ \left( 
\begin{array}{c}
y_{1} \\
y_{2} \\
y_{3} \\
y_{4} \\
y_{5} \\
y_{6} 
 \end{array}
\right)
 = \left( 
\begin{array}{cccccc}
1 & u_{{1}{2}} & u_{{1}{3}} & u_{{1}{4}} & u_{{1}{5}} & u_{{1}{6}} \\
0 & 1 & u_{{2}{3}} & u_{{2}{4}} & u_{{2}{5}} & u_{{2}{6}} \\
0 & 0 & 1 & u_{{3}{4}} & u_{{3}{5}} & u_{{3}{6}} \\
0 & 0 & 0 & 1 & u_{{4}{5}} & u_{{4}{6}} \\
0 & 0 & 0 & 0 & 1 & u_{{5}{6}} \\
0 & 0 & 0 & 0 & 0 & 1 
 \end{array}
\right)
 \cdot \left( 
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3} \\
x_{4} \\
x_{5} \\
x_{6} 
 \end{array}
\right)
 $$
\begin{multicols}{2}
$$ y_{6} = x_{6} $$
$$ y_{5} = x_{5} + u_{{5}{6}} x_{6} $$
$$ y_{4} = x_{4} + u_{{4}{5}} x_{5} + u_{{4}{6}} x_{6} $$
$$ y_{3} = x_{3} + u_{{3}{4}} x_{4} + u_{{3}{5}} x_{5} + u_{{3}{6}} x_{6} $$
$$ y_{2} = x_{2} + u_{{2}{3}} x_{3} + u_{{2}{4}} x_{4} + u_{{2}{5}} x_{5} + u_{{2}{6}} x_{6} $$
$$ y_{1} = x_{1} + u_{{1}{2}} x_{2} + u_{{1}{3}} x_{3} + u_{{1}{4}} x_{4} + u_{{1}{5}} x_{5} + u_{{1}{6}} x_{6} $$
\vfill\null
\columnbreak
$$ x_{6} = y_{6} $$
$$ x_{5} = y_{5}- u_{{5}{6}} x_{6} $$
$$ x_{4} = y_{4}- u_{{4}{5}} x_{5}- u_{{4}{6}} x_{6} $$
$$ x_{3} = y_{3}- u_{{3}{4}} x_{4}- u_{{3}{5}} x_{5}- u_{{3}{6}} x_{6} $$
$$ x_{2} = y_{2}- u_{{2}{3}} x_{3}- u_{{2}{4}} x_{4}- u_{{2}{5}} x_{5}- u_{{2}{6}} x_{6} $$
$$ x_{1} = y_{1}- u_{{1}{2}} x_{2}- u_{{1}{3}} x_{3}- u_{{1}{4}} x_{4}- u_{{1}{5}} x_{5}- u_{{1}{6}} x_{6} $$
\end{multicols}
\end{document}