## DESCRIPTION ## Linear Algebra ## ENDDESCRIPTION ## KEYWORDS ('linear algebra','matrix','eigenvalue','characteristic polynomial') ## Tagged by cmd6a 4/30/06 ## DBsubject('Linear Algebra') ## DBchapter('Matrices') ## DBsection('Eigenvalues') ## Date('March 14, 2010') ## Author('edited by Shifrin') ## Institution('UGA') ## 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", "MathObjects.pl" ); TEXT(beginproblem()); \$showPartialCorrectAnswers = 1; Context("Matrix")->variables->are(t=>'Real'); \$a11 = non_zero_random(-5,5,1); \$a12 = non_zero_random(-5,5,1); \$a13 = 0; \$a21 = 0; \$a22 = non_zero_random(-5,5,1); \$a23 = non_zero_random(-5,5,1); \$a31 = non_zero_random(-5,5,1); \$a32 = non_zero_random(-5,5,1); \$a33 = 0; \$A = Matrix("[[\$a11,\$a12,\$a13],[\$a21,\$a22,\$a23],[\$a31,\$a32,\$a33]]"); \$k = \$a11 + \$a22; \$m = \$a23 * \$a32 - \$a11 * \$a22; \$n = \$a12 * \$a23 * \$a31 - \$a11 * \$a23 * \$a32; \$f = Formula("-t^3 + \$k t^2 + \$m t + \$n")->reduce->cmp; Context()->texStrings; BEGIN_TEXT The characteristic polynomial of the matrix \$PAR \( A = \$A\) \$PAR is \( p(t) = \) \{ans_rule(40)\}. END_TEXT ANS(\$f); ENDDOCUMENT() ;