## DESCRIPTION ## Linear Algebra ## ENDDESCRIPTION ## KEYWORDS ('linear algebra','matrix','skew-symmetric') ## Tagged by cmd6a 5/3/06 ## DBsubject('Linear Algebra') ## DBchapter('Matrices') ## DBsection('Diagonalization') ## Date('') ## Author('') ## Institution('Rochester') ## TitleText1('') ## EditionText1('') ## AuthorText1('') ## Section1('') ## Problem1('') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGgraphmacros.pl", "PGmatrixmacros.pl", "PGnumericalmacros.pl", "PGauxiliaryFunctions.pl", "PGmorematrixmacros.pl" ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1; @a = NchooseK(6,6);$i1 = random(1,2,1); $j1 = 3 -$i1; $i2 = random(1,3,2);$j2 = 4 - $i2;$i3 = random(2,3,1); $j3 = 5 -$i3; $a[1] = non_zero_random(-5,5,1);$a[2] = $a[1]+non_zero_random(-3,3,2);$a[3] = $a[1]+non_zero_random(-4,4,2); BEGIN_TEXT Enter a $$3 \times 3$$ skew-symmetric matrix $$A$$ that has entries$BR $$a_{i1 j1}=a[1]$$, $$a_{i2 j2}=a[2]$$, $$a_{i3 j3}=a[3]$$. $BR \{ mbox( '$$A=$$', answer_matrix(3,3,5), '.' ) \} END_TEXT if ($i1==1) { $a[12] =$a[1]; $a[21] = -$a[1]; } else { $a[12] = -$a[1]; $a[21] =$a[1]; } if ($i2==1) {$a[13] = $a[2];$a[31] = - $a[2]; } else {$a[13] = - $a[2];$a[31] = $a[2]; } if ($i3==2) { $a[23] =$a[3]; $a[32] = -$a[3]; } else { $a[23] = -$a[3]; $a[32] =$a[3]; } ANS(num_cmp(0)); ANS(num_cmp($a[12])); ANS(num_cmp($a[13])); ANS(num_cmp($a[21])); ANS(num_cmp(0)); ANS(num_cmp($a[23])); ANS(num_cmp($a[31])); ANS(num_cmp($a[32])); ANS(num_cmp(0)); ENDDOCUMENT(); # This should be the last executable line in the problem.