#Title: Sistema de três lineares caso I

#Let:

n x1 = [ 0 , 1  , 2 , -1 , -3 , 5 , 3] ;
n x2 = [ 1 , 2 , 3 , -2 , -1 , -3 , 4] ;
n x3 = [ 1 , 2 , 3 , -2 , -1 , -3 , 4] ;

n a11 = [1 , 2 , 3 , -2 , -1 , -3 , 4] ;
n a12 = [1 , 2 , 3 , -2 , -1 , -3 , 4] ;
n a13 = [1 , 2 , 3 , -2 , -1 , -3 , 4] ;

n a22p = [1 , 2 , 3 , -2 , -1 , -3 , 4] ;
n a23p = [1 , 2 , 3 , -2 , -1 , -3 , 4] ;

n a33pp = [1 , 2 , 3 , -2 , -1 , -3 , 4] ;

n m21 = [1 , 2 , 3 , -2 , -1 , -3 , 4] ;
n m31 = [1 , 2 , 3 , -2 , -1 , -3 , 4] ;
n m32 = [1 , 2 , 3 , -2 , -1 , -3 , 4] ;

cn a21 = #m21 * #a11 ;
cn a31 = #m31 * #a11 ;
cn a32p = #m32 * #a22p ;
cn a22 = #a22p + #m21 * #a12 ;
cn a23 = #a23p + #m21 * #a13 ;
cn a33p = #a33pp + #m32 * #a23p ;
cn a33 = #a33p + #m31 * #a13 ;
cn a32 = #a32p + #m31 * #a12 ;

cn b1 = #a11 * #x1 + #a12 * #x2 + #a13 * #x3 ;
cn b2 = #a21 * #x1 + #a22 * #x2 + #a23 * #x3 ;
cn b3 = #a31 * #x1 + #a32 * #x2 + #a33 * #x3 ;

cn b2p = #b2 - #m21 * #b1 ;
cn b3p = #b3 - #m31 * #b1 ;
cn b3pp = #b3p - #m32 * #b2p ;

#Question:

\noindent Seja $(S)$ o sistema de equações lineares cuja matriz dos coeficientes
é $A=\smleft #a11 & #a12 & #a13 \\ #a21 & #a22 & #a23 \\ #a31 & #a32 & #a33 \smright$ e o vector
dos termos independentes é $b=\smleft #b1 \\ #b2 \\ #b3 \smright$. Determine $\CS_{(S)}$.

#Sugestion:

\noindent Utilize o Método de Gauss.


#resolution:

$
\left[
\begin{array}{ccc|c}
\fbox{#a11}  & #a12 & #a13 & #b1 \\
 #a21 & #a22 & #a23 & #b2 \\
 #a31 & #a32 & #a33 & #b3
\end{array}
\right]
\begin{array}{l}
\xlongleftrightarrow{\hspace{2.2cm}}\\
\ell_2\leftarrow\ell_2-(#m21)\ell_1\\
\ell_3\leftarrow\ell_3-(#m31)\ell_1\\
\end{array}
\left[
\begin{array}{ccc|c}
 #a11  & #a12 & #a13 & #b1 \\
0 & \fbox{#a22p} & #a23p & #b2p \\
0 & #a32p & #a33p & #b3p
\end{array}
\right]
\begin{array}{l}
\xlongleftrightarrow{\hspace{2.2cm}}\\
\\
\ell_3\leftarrow\ell_3-(#m32)\ell_2\\
\end{array}
\left[
\begin{array}{ccc|c}
 #a11  & #a12 & #a13 & #b1 \\
0 & #a22p & #a23p & #b2p \\
0 & 0 & #a33pp & #b3pp
\end{array}
\right]
$

Assim, tem-se:

\begin{gather*}
x_3=\frac{#b3pp}{#a33pp}=#x3\\
x_2=\frac{#b2p-#a23p\times#x3}{#a22p}=#x2\\
x_1=\frac{#b1-#a12\times#x2-#a13\times#x3}{#a11}=#x1
\end{gather*}

#Result:

$CS_{(S)}=\{(#x1,#x2,#x3)\}$


#Verification:

vector(#x1, #x2, #x3) ;

#usepackage

\usepackage{amsmath,amssymb}
\usepackage{extarrows}
\makeatletter
% smallmatrix with right alignment
\newenvironment{rsmallmatrix}{\null\,\vcenter\bgroup
  \Let@\restore@math@cr\default@tag
  \baselineskip6\ex@ \lineskip1.5\ex@ \lineskiplimit\lineskip
  \ialign\bgroup\hfil$\m@th\scriptstyle##$&&\thickspace\hfil
  $\m@th\scriptstyle##$\crcr
}{%
  \crcr\egroup\egroup\,%
}
\makeatother

\newcommand{\smleft}{\left[\begin{rsmallmatrix}}
\newcommand{\smright}{\end{rsmallmatrix}\right]}
\DeclareMathOperator{\CS}{CS}