%This command provides the cramers rule in the column 1
%of the third horizontal part of page 10
%
%This command has one parameter:
%      1) The width of the line used to typeset the formulae
\newcommand\TTenCramer[1]{%
   \parbox[t]{#1}{%
      \TTenCramerFontSize
      \DisplaySpace{\TTenDisplaySpace}{\TTenDisplayShortSpace}
      \noindent If we have equations:%
      \[\begin{array}{l%Col 2 plus sign
                    @{\hspace{.2em}}c@{\hspace{.2em}}% Col 3
                    l%Col 4, \cdot
                    wc{4em}%Col 5, plus sign
                    @{\hspace{.2em}}c@{\hspace{.2em}}% Col 6
                    l%Col 7 equal sign
                    @{\hspace{.2em}}c@{\hspace{.2em}}% Col 8
                    l}
       
         a_{1,1} x_1 &+& a_{1,2} x_2& \makebox[2em][c]{$\cdots$} &+& a_{1,n} x_n &=& b_1 \\
         a_{2,1} x_1 &+& a_{2,2} x_2& \makebox[2em][c]{$\cdots$} &+& a_{2,n} x_n &=& b_2 \\
                     & &            &                            & &             & &     \\
                     & &            & \vdots                     & &             & &     \\ 
                     & &            &                            & &             & &     \\
         a_{n,1} x_1 &+& a_{n,2} x_2& \makebox[2em][c]{$\cdots$} &+& a_{n,n} x_n &=& b_n \\
      \end{array}\]

      \AdjustSpace{3ex plus .5 ex minus 1ex}

      Let $A = (a_{i,j})$ and $B$ be the column matrix $(b_i)$.
      Then there is a unique solution iff $\det A \neq 0$.
      Let $A_i$ be $A$ with column $i$ replaced by $B$.
      Then
      \begin{displaymath}
         x_i = \frac{\det A_i}{\det A}.
      \end{displaymath}
   }%
}
%The command containing the title of this part
\newcommand\TTenCramersTitle{Cramer's rule}