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| 1 : | ted shifri | 1457 | ## DESCRIPTION |
| 2 : | ## Linear Algebra | ||
| 3 : | ## ENDDESCRIPTION | ||
| 4 : | |||
| 5 : | ## KEYWORDS ('linear algebra','matrix','skew-symmetric') | ||
| 6 : | ## Tagged by cmd6a 5/3/06 | ||
| 7 : | |||
| 8 : | ## DBsubject('Linear Algebra') | ||
| 9 : | ## DBchapter('Matrices') | ||
| 10 : | ## DBsection('Diagonalization') | ||
| 11 : | ## Date('') | ||
| 12 : | ## Author('') | ||
| 13 : | ## Institution('Rochester') | ||
| 14 : | ## TitleText1('') | ||
| 15 : | ## EditionText1('') | ||
| 16 : | ## AuthorText1('') | ||
| 17 : | ## Section1('') | ||
| 18 : | ## Problem1('') | ||
| 19 : | |||
| 20 : | DOCUMENT(); # This should be the first executable line in the problem. | ||
| 21 : | |||
| 22 : | loadMacros( | ||
| 23 : | "PG.pl", | ||
| 24 : | "PGbasicmacros.pl", | ||
| 25 : | "PGchoicemacros.pl", | ||
| 26 : | "PGanswermacros.pl", | ||
| 27 : | "PGgraphmacros.pl", | ||
| 28 : | "PGmatrixmacros.pl", | ||
| 29 : | "PGnumericalmacros.pl", | ||
| 30 : | "PGauxiliaryFunctions.pl", | ||
| 31 : | "PGmorematrixmacros.pl" | ||
| 32 : | ); | ||
| 33 : | |||
| 34 : | TEXT(beginproblem()); | ||
| 35 : | $showPartialCorrectAnswers = 1; | ||
| 36 : | |||
| 37 : | @a = NchooseK(6,6); | ||
| 38 : | $i1 = random(1,2,1); | ||
| 39 : | $j1 = 3 - $i1; | ||
| 40 : | $i2 = random(1,3,2); | ||
| 41 : | $j2 = 4 - $i2; | ||
| 42 : | $i3 = random(2,3,1); | ||
| 43 : | $j3 = 5 - $i3; | ||
| 44 : | $a[1] = non_zero_random(-5,5,1); | ||
| 45 : | $a[2] = $a[1]+non_zero_random(-3,3,2); | ||
| 46 : | $a[3] = $a[1]+non_zero_random(-4,4,2); | ||
| 47 : | |||
| 48 : | |||
| 49 : | BEGIN_TEXT | ||
| 50 : | |||
| 51 : | Enter a \( 3 \times 3 \) skew-symmetric matrix \(A\) that has entries | ||
| 52 : | $BR \(a_{$i1 $j1}=$a[1]\), \(a_{$i2 $j2}=$a[2]\), \(a_{$i3 $j3}=$a[3]\). | ||
| 53 : | $BR | ||
| 54 : | \{ mbox( '\(A=\)', answer_matrix(3,3,5), '.' ) \} | ||
| 55 : | |||
| 56 : | END_TEXT | ||
| 57 : | |||
| 58 : | if ($i1==1) { | ||
| 59 : | $a[12] = $a[1]; | ||
| 60 : | $a[21] = - $a[1]; | ||
| 61 : | } else { | ||
| 62 : | $a[12] = - $a[1]; | ||
| 63 : | $a[21] = $a[1]; | ||
| 64 : | } | ||
| 65 : | |||
| 66 : | if ($i2==1) { | ||
| 67 : | $a[13] = $a[2]; | ||
| 68 : | $a[31] = - $a[2]; | ||
| 69 : | } else { | ||
| 70 : | $a[13] = - $a[2]; | ||
| 71 : | $a[31] = $a[2]; | ||
| 72 : | } | ||
| 73 : | |||
| 74 : | if ($i3==2) { | ||
| 75 : | $a[23] = $a[3]; | ||
| 76 : | $a[32] = - $a[3]; | ||
| 77 : | } else { | ||
| 78 : | $a[23] = - $a[3]; | ||
| 79 : | $a[32] = $a[3]; | ||
| 80 : | } | ||
| 81 : | |||
| 82 : | ANS(num_cmp(0)); ANS(num_cmp($a[12])); ANS(num_cmp($a[13])); | ||
| 83 : | ANS(num_cmp($a[21])); ANS(num_cmp(0)); ANS(num_cmp($a[23])); | ||
| 84 : | ANS(num_cmp($a[31])); ANS(num_cmp($a[32])); ANS(num_cmp(0)); | ||
| 85 : | |||
| 86 : | ENDDOCUMENT(); # This should be the last executable line in the problem. | ||
| 87 : |
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