## DESCRIPTION
## Linear Algebra
## ENDDESCRIPTION

## KEYWORDS ('linear algebra','matrix','eigenvalue')
## Tagged by cmd6a 4/30/06

## DBsubject('Linear Algebra')
## DBchapter('Matrices')
## DBsection('Eigenvalues')
## 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 = non_zero_random(-2,2,1);
$e = random(1,2,1) + 3*random(-1,1,1) - $a;
$b = random(-1,1,2);
$sqrD = random(1,2,1) + 3*random(0,2,1);
$D = $sqrD**2;
$d = ($D - ($a+$e)**2)*$b/3 + $a*$e*$b; 

# characteristic polynomial is - (lambda^3 - (a+e)lambda^2 + (ae-bd)lambda - b^2k) 
# now forget about the minus in front
# we want 3 distinct real roots, so we want local max and min with pos and neg values respectively
# find max and min: 
# derivative is 3lamda^2 - 2(a+e)lambda + (ae-bd)
# need 2 disctinct real roots
# discriminant/4 is (a+e)^2 - 3ae + 3bd = (a+e)^2 - 3ae + D - (a+e)^2 + 3ae = D
# ok, the discriminant is positive
# roots are  

$max_root = ($a + $e + $sqrD)/3;
$min_root = ($a + $e - $sqrD)/3;

# local max and min values of the cubic polynomial (without the minus in front) have to be pos and neg, so 

$ans_min = $max_root**3 - ($a+$e)*$max_root**2 + ($a*$e - $b*$d)*$max_root;
$ans_max = $min_root**3 - ($a+$e)*$min_root**2 + ($a*$e - $b*$d)*$min_root;

BEGIN_TEXT

\{ mbox( 'The matrix \(A=\)', display_matrix([[$a, $b, 0], [$d, $e, $b], ['k', 0, 0]]) )\} 
$BR
has three distinct real eigenvalues if and only if 
$BR
\{ans_rule(20)\} \( < k < \) \{ans_rule(20)\}. 

END_TEXT

ANS(num_cmp($ans_min));
ANS(num_cmp($ans_max));

ENDDOCUMENT();       # This should be the last executable line in the problem.