## 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.