View of /branches/UGA/9.4.1.pg

    1 ## DESCRIPTION
    2 ## Linear Algebra
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS ('linear algebra','eigenvalues', 'diagonalizability')
    6 
    7 ## DBsubject('Multivariable Mathematics')
    8 ## DBchapter('Linear Transformations')
    9 ## DBsection('Eigenvalues and Eigenvectors, Diagonalizability')
   10 ## Date('March 13, 2010')
   11 ## Author('Shifrin')
   12 ## Institution('UGA')
   13 ## TitleText1('')
   14 ## EditionText1('')
   15 ## AuthorText1('')
   16 ## Section1('')
   17 ## Problem1('')
   18 
   19 DOCUMENT();        # This should be the first executable line in the problem.
   20 
   21 loadMacros("PG.pl",
   22            "PGbasicmacros.pl",
   23            "PGchoicemacros.pl",
   24            "PGanswermacros.pl",
   25            "PGauxiliaryFunctions.pl",
   26            "PGmatrixmacros.pl",
   27            "PGmorematrixmacros.pl",
   28            "Parser.pl",
   29            "weightedGrader.pl"
   30            );
   31 
   32 
   33 TEXT(beginproblem());
   34 $showPartialCorrectAnswers = 0;
   35 
   36 install_weighted_grader();
   37 
   38 Context("Matrix");
   39 
   40 $a=random(-3,3);
   41 $b=non_zero_random(-3,3);
   42 do{$c=random(-3,3)} until ($c!=$a);
   43 do{$d=random(-3,3)} until ($d!=$a && $d!=$c);
   44 do{$e=random(-3,3)} until ($e!=$a && $e!=$c && $e!=$d);
   45 $r1=random(-5,5);
   46 $r2=random(-5,5);
   47 $r3=random(-5,5);
   48 $r4=non_zero_random(-5,5);
   49 $r5=random(-5,5);
   50 $r6=random(-5,5);
   51 
   52 
   53 $M1 = Matrix("[[$a,$b,$r1,$r2],[0,$a,$r3,$r4],[0,0,$c,$r5],[0,0,0,$d]]");
   54 $M2 = Matrix("[[$a,0,0,0],[$r1,$c,0,0],[$r2,$r3,$d,0],[$r4,$r5,$r2,$e]]");
   55 $M3 = Matrix("[[$a,$r1,0,$r2],[0,$c,0,0],[0,$r4,$a,$r5],[0,0,0,$c]]");
   56 $M4 = Matrix("[[$a,$r3,$r2,$r1],[$r3,$a,$r5,$r4],[$r2,$r5,$c,$r6],
   57 [$r1,$r4,$r6,$d]]");
   58 
   59 
   60 @Ms = ($M1,$M2,$M3,$M4);
   61 @diags=('not diagonalizable','diagonalizable','diagonalizable','diagonalizable');
   62 
   63 @choice=NchooseK(4,4);
   64 @A=@Ms[@choice];
   65 @diag=@diags[@choice];
   66 
   67 $ans1=str_cmp($diag[0]);
   68 $ans2=str_cmp($diag[1]);
   69 $ans3=str_cmp($diag[2]);
   70 $ans4=str_cmp($diag[3]);
   71 
   72 
   73 
   74 Context()->texStrings;
   75 BEGIN_TEXT
   76 
   77 Consider the matrices
   78 \[A = $A[0], \quad B = $A[1], \]
   79 $BR
   80 \[C = $A[2], \quad D = $A[3] . \]
   81 
   82 $PAR
   83 The matrix \(A\) is \{pop_up_list(['?','diagonalizable','not diagonalizable'])\}.
   84 $PAR
   85 The matrix \(B\) is \{pop_up_list(['?','diagonalizable','not diagonalizable'])\}.
   86 $PAR
   87 The matrix \(C\) is \{pop_up_list(['?','diagonalizable','not diagonalizable'])\}.
   88 $PAR
   89 The matrix \(D\) is \{pop_up_list(['?','diagonalizable','not diagonalizable'])\}.
   90 $PAR
   91 
   92 END_TEXT
   93 
   94 ANS($ans1,$ans2,$ans3,$ans4);
   95 
   96 ENDDOCUMENT();       # This should be the last executable line in the problem.
   97 
   98