## 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() ;