## DESCRIPTION
##   Constraint equation for consistency
## ENDDESCRIPTION

## KEYWORDS('Linear equations')
## 

## DBsubject('Calculus')
## DBchapter('')
## DBsection('')
## Date('10/04/2009')
## Author('Ted Shifrin')
## Institution('UGA')
## TitleText1('')
## EditionText1('')
## AuthorText1('')
## Section1('')
## Problem1('')
           
DOCUMENT();	# This should be the first executable line in the problem.

loadMacros("PGstandard.pl",      
           "PGbasicmacros.pl",
           "PGchoicemacros.pl",      
           "PGanswermacros.pl",
           "PGauxiliaryFunctions.pl",
           "PGmatrixmacros.pl",
           "PGmorematrixmacros.pl",
          );      
         
     
TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = non_zero_random(-2, 2, 1);
$b = non_zero_random(-2, 2, 1);
$b1 = non_zero_random(-2, 2, 1);
$c = non_zero_random(-2, 2, 1);
$d = non_zero_random(-2, 2, 1);
$d1 = non_zero_random(-2, 2, 1);
$e = non_zero_random(-3, 3, 1);
$f = non_zero_random(-3, 3, 1);
$i = non_zero_random(-1, 1, 2);
$g = $e*$f+$i;
$k = non_zero_random(-3, 3, 1);
$l = non_zero_random(-3, 3, 1);
$m = non_zero_random(-3, 3, 1);
$n = non_zero_random(-3, 3, 1);


$m11=1; $m12=$e; $m13=$a+$e*$c; $m14=$b+$e*$d; $m15=$b1+$e*$d1;
$m21=$f; $m22=$g; $m23=$f*$a+$g*$c; $m24=$f*$b+$g*$d; $m25=$f*$b1+$g*$d1;
$m31=$k; $m32=$l; $m33=$k*$a+$l*$c; $m34=$k*$b+$l*$d; $m35=$k*$b1+$l*$d1;
$m41=$m; $m42=$n; $m43=$m*$a+$n*$c; $m44=$m*$b+$n*$d; $m45=$m*$b1+$n*$d1;

$ans11=-$k+$i*$f*($l-$e*$k);
$ans12=-$i*($l-$e*$k);
$ans13=1;
$ans14=0;

$ans21=-$m+$i*$f*($n-$e*$m);
$ans22=-$i*($n-$e*$m);
$ans23=0;
$ans24=1;

BEGIN_TEXT

Find the constraint equations for \(A\mathbf x = \mathbf b\) to be consistent, with

\[ A = \left[\begin{array}{r r r r r} 
$m11 & $m12 & $m13 & $m14 & $m15 \cr
$m31 & $m32 & $m33 & $m34 & $m35 \cr
$m21 & $m22 & $m23 & $m24 & $m25 \cr
$m41 & $m42 & $m43 & $m44 & $m45  
\end{array} \right]\quad .
\]

$PAR

Constraint equations are

\{ mbox( ans_array(4,1,8), '\(\cdot \ \mathbf b = \quad\)',   ans_array_extension(4,1,8), '\(\cdot \ \mathbf b = 0\ .\)') \}


END_TEXT

ANS(basis_cmp([[$ans11,$ans13,$ans12,$ans14],[$ans21,$ans23,$ans22,$ans24]]));



ENDDOCUMENT();	# This should be the last executable line in the problem.