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```    1 ## DESCRIPTION
2 ## Linear Algebra
3 ## ENDDESCRIPTION
4
5 ## KEYWORDS ('linear algebra','matrix','eigenvalue')
6 ## Tagged by cmd6a 4/30/06
7
8 ## DBsubject('Linear Algebra')
9 ## DBchapter('Matrices')
10 ## DBsection('Eigenvalues')
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 = non_zero_random(-8,8,1);
38 \$b = non_zero_random(-9,9,1);
39 \$c = non_zero_random(-8,8,1);
40 if (\$a == \$c) { \$c=9; }
41
42 \$k = - (\$a - \$c)**2/(4*\$b);
43
44 BEGIN_TEXT
45
46 For which value of \(k\) does the matrix
47 \$BR
48 \{ mbox( '\(A=\)', display_matrix([[\$a, 'k'], [\$b, \$c]]) )\}
49 \$BR
50 have one real eigenvalue of multiplicity \(2\)?
51 \$BR
52 \(k=\) \{ans_rule(20)\}.
53
54 END_TEXT
55
56 ANS(num_cmp(\$k));
57
58 ENDDOCUMENT();       # This should be the last executable line in the problem.
59
```

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