%=========================================================================%
% initialize TeX
%=========================================================================%

\magnification=\magstep1
\baselineskip=14pt
\hsize = 6.5truein
\hoffset = 0.0truein
\vsize = 9.1truein
\voffset = -0.25truein
\emergencystretch = 0.05\hsize

\newif\ifblackboardbold

% comment out the following line if AMS msbm fonts aren't available
\blackboardboldtrue

%=========================================================================%
% select fonts
%=========================================================================%

\font\titlefont=cmbx12 scaled\magstephalf
\font\sectionfont=cmbx12
\font\netfont=cmtt9

% Establish AMS blackboard bold fonts without using amssym.def, amssym.tex

\newfam\bboldfam
\ifblackboardbold
\font\tenbbold=msbm10
\font\sevenbbold=msbm7
\font\fivebbold=msbm5
\textfont\bboldfam=\tenbbold
\scriptfont\bboldfam=\sevenbbold
\scriptscriptfont\bboldfam=\fivebbold
\def\bbold{\fam\bboldfam\tenbbold}
\else
\def\bbold{\bf}
\fi

%=========================================================================%
% exam macros (use \practice or \exam to initialize)
%=========================================================================%

\newcount\probcount
\probcount=0

\def\practice{
\def\prob##1\par
{\medskip\global\advance\probcount by 1\noindent
{\bf[\number\probcount]}~##1}}

\def\exam{
\def\prob##1\par
{\global\advance\probcount by 1\noindent {\bf[\number\probcount]}~##1
\vfill\eject
\noindent{\bf Problem: \hbox to 30pt{\hrulefill}}
\vfill\eject
}
\footline={\tenbf Page \folio\hfil Continued on page: \hbox to
30pt{\hrulefill}}
% set up name, id, school fields for exam
\vskip 0.1in 
{\bf \noindent Name: \hrulefill \hbox{}}
\vskip 0.1in 
{\bf \noindent ID: \hrulefill ~ School: \hrulefill \hbox{}}
\vskip 0.2in
}

% set up those boxes to put the scores in

\def\boxAsub#1{\vbox{\hbox{#1}}}
\def\boxA#1#2{[{\bf #1}] (#2 pts)}
\def\boxT{\bf TOTAL}
\def\boxB{\vbox to 25pt{\hbox{\kern 56pt}}}

\def\fiveboxes#1{
\centerline{
\vbox{\offinterlineskip\halign{
 \strut\vrule\hfil##\hfil\vrule
 &&\hfil##\hfil\vrule\cr
 \noalign{\hrule}
 \boxA 1{#1} &
 \boxA 2{#1} &
 \boxA 3{#1} &
 \boxA 4{#1} &
 \boxA 5{#1} &
 \boxT \cr
 \noalign{\hrule}
 \boxB & \boxB & \boxB & \boxB & \boxB & \boxB  \cr
 \noalign{\hrule}
}}}}

\def\eightboxes#1{
\centerline{
\vbox{\offinterlineskip\halign{
 \strut\vrule\hfil##\hfil\vrule
 &&\hfil##\hfil\vrule\cr
 \multispan4{\hrulefill}\cr
 \boxA 1{#1} &
 \boxA 2{#1} &
 \boxA 3{#1} &
 \boxA 4{#1} \cr
 \multispan4{\hrulefill}\cr
 \boxB & \boxB & \boxB & \boxB \cr
 \noalign{\hrule}
 \boxA 5{#1} &
 \boxA 6{#1} &
 \boxA 7{#1} &
 \boxA 8{#1} &
 \boxT \cr
 \noalign{\hrule}
 \boxB & \boxB & \boxB & \boxB & \boxB\cr
 \noalign{\hrule}
}}}}

\def\tenboxes#1{
\centerline{
\vbox{\offinterlineskip\halign{
 \strut\vrule\hfil##\hfil\vrule
 &&\hfil##\hfil\vrule\cr
 \multispan5{\hrulefill}\cr
 \boxA 1{#1} &
 \boxA 2{#1} &
 \boxA 3{#1} &
 \boxA 4{#1} &
 \boxA 5{#1} \cr
 \multispan5{\hrulefill}\cr
 \boxB & \boxB & \boxB & \boxB & \boxB \cr
 \noalign{\hrule}
 \boxA 6{#1} &
 \boxA 7{#1} &
 \boxA 8{#1} &
 \boxA 9{#1} &
 \boxA{10}{#1} &
 \boxT \cr
 \noalign{\hrule}
 \boxB & \boxB & \boxB & \boxB & \boxB & \boxB\cr
 \noalign{\hrule}
}}}}

%==========================================================================
% macros
%==========================================================================

\def\bmatrix#1{{\left[\vcenter{\halign
  {&\kern 4pt\hfil$##\mathstrut$\kern 4pt\cr#1}}\right]}}

% surround with $ $ if not already in math mode
\def\enma#1{{\ifmmode#1\else$#1$\fi}}

\def\mathbb#1{{\bbold #1}}
\def\mathbf#1{{\bf #1}}

% blackboard bold symbols
\def\FF{\enma{\mathbb{F}}}
\def\NN{\enma{\mathbb{N}}}
\def\RR{\enma{\mathbb{R}}}
\def\CC{\enma{\mathbb{C}}}
\def\QQ{\enma{\mathbb{Q}}}
\def\ZZ{\enma{\mathbb{Z}}}
\def\bbPP{\enma{\mathbb{P}}}

% caligraphic symbols
\def\cAA{\enma{\cal A}}
\def\cBB{\enma{\cal B}}

% bold symbols
\def\aa{\enma{\mathbf{a}}}
\def\bb{\enma{\mathbf{b}}}
\def\cc{\enma{\mathbf{c}}}
\def\ee{\enma{\mathbf{e}}}
\def\mm{\enma{\mathbf{m}}}
\def\pp{\enma{\mathbf{p}}}
\def\vv{\enma{\mathbf{v}}}
\def\ww{\enma{\mathbf{w}}}
\def\xx{\enma{\mathbf{x}}}
\def\yy{\enma{\mathbf{y}}}
\def\XX{\enma{\mathbf{X}}}
\def\KK{\enma{\mathbf{K}}}
\def\PP{\enma{\mathbf{P}}}
\def\LL{\enma{\mathbf{L}}}
\def\II{\enma{\mathbf{I}}}
\def\zero{\enma{\mathbf{0}}}
\def\blambda{\enma{\mathbf{\lambda}}}

% misc

\def\abs#1{\enma{\left| #1 \right|}}
\def\set#1{\enma{\{#1\}}}
\def\setdef#1#2{\enma{\{\;#1\;\,|\allowbreak
  \;\,#2\;\}}}
\def\idealdef#1#2{\enma{\langle\;#1\;\,
  |\;\,#2\;\rangle}}

%=========================================================================%
% exam
%=========================================================================%

\centerline{\titlefont Exam 1}
\smallskip
\centerline{Linear Algebra, Dave Bayer, October 7, 1999}
\medskip

%\practice
\exam
\fiveboxes{6}

Please work only one problem per page, starting with the pages provided, and
number all continuations clearly. Only work which can be found in this way will
be graded.

Please do not use calculators or decimal notation.

\bigskip

\prob
Solve the following system of equations:
$$
\bmatrix
{0 & 1 & 1 \cr
1 & 0 & 1 \cr
1 & 1 & 0 \cr}
\bmatrix{x \cr y \cr z \cr}
=
\bmatrix{2 \cr 0 \cr 0 \cr}
$$

\prob
Compute matrices giving the number of walks of lengths 1, 2, and 3
between pairs of vertices of the following graph:
\input epsf
$$\epsfbox{graph2.eps}$$

\prob
Express the following matrix as a product of elementary matrices:
$$\bmatrix{
0 & 1 & 1 \cr
1 & 0 & 1 \cr
1 & 1 & 0 \cr
}$$


\prob
Compute the determinant of the following $4\times 4$ matrix:
$$\bmatrix{
1 & 1 & 1 & 0 \cr
2 & 2 & 0 & 2 \cr
3 & 0 & 3 & 3 \cr
0 & 4 & 4 & 4 \cr
}$$
What can you say about the determinant of the $n\times n$ matrix with the same
pattern?


\prob
Use Cramer's rule to give a formula for the solution to the following
system of equations:
$$
\bmatrix
{0 & 1 & 1 \cr
1 & 0 & 1 \cr
1 & 1 & 0 \cr}
\bmatrix{x \cr y \cr z \cr}
=
\bmatrix{2a \cr 2b \cr 2c\cr}
$$

\bye