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\begin{document}

\centerline{\bf MAT 2116 - \'Algebra Linear para Qu\'imica} 

\centerline{\bf 1$^a$ Prova - 20 de abril de 2017}


\noindent
    {\bf Quest\ao\ 1} (2,5 pts)
    \\
    (a) Determine a matriz dos coeficientes de um sistema linear homog\^eneo que seja satisfeito pelos
    valores de $x$, $y$, $z$ e $w$ que equilibrem a equa\cao\ qu\'imica
    \begin{center}
    $
   x\,\mbox{N}_2\mbox{H}_4\ +\ y\,\mbox{N}_2\mbox{O}_4\ \ \longrightarrow\ \
     z\,\mbox{N}_2\ +\ w\,\mbox{H}_2\mbox{O}
    $.
    \end{center}
(b) Determine, se houver, uma solu\cao\ inteira com $z=27$ do sistema encontrado no item (a). 



\noindent
    {\bf Quest\ao\ 2} (2,5 pts) Determine o determinante e a inversa da matriz dada.

    (a) ${\displaystyle A=\left[\begin{array}{cccccc}
        0 & 0 & 0 & 1 & 0 & 0  \\
        0 & 1 & 0 & 0 & 0 & 0  \\
        0 & 0 & 1 & 0 & 0 & 0  \\
        1 & 0 & 0 & 0 & 0 & 0  \\
        0 & 0 & 0 & 0 & 1 & 0  \\
        0 & 0 & 0 & 0 & 0 & 1
        \end{array}\right] }$
\ \ \ \ \ \ \ \ \ \ 
    (b)  ${\displaystyle B=\left[\begin{array}{cccccc}
        1 & 0 & 0 & 0 & 0 & 0  \\
        0 & 1 & 0 & 0 & 0 & 9  \\
        0 & 0 & 1 & 0 & 0 & 0  \\
        0 & 0 & 0 & 1 & 0 & 0  \\
        0 & 0 & 0 & 0 & 1 & 0  \\
        0 & 0 & 0 & 0 & 0 & 1
        \end{array}\right] }$




\noindent
    {\bf Quest\ao\ 3} (2,5 pts) Sendo  ${\displaystyle A=\left[\begin{array}{cc}
2 & 3 \\ 3 & 2
        \end{array}\right] }$, escreva $A^{-1}$ como produto de matrizes elementares.

    



\noindent
{\bf Quest\ao\ 4} (2,5 pts)  Sendo  ${\displaystyle A=\left[\begin{array}{ccc}
3 & 0 & -2 \\ 0 & 1 & 0 \\ 5 & 0 & 5
    \end{array}\right] }$, determine os valores de $t\in\R$ para os quais
$A-tI$ n\ao\ \'e invers\'ivel.




\end{document}
