View of /trunk/NationalProblemLibrary/ASU-topics/set119MatrixAlgebra/p13.pg

    1 ## DESCRIPTION
    2 ## Matrix Algebra
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('Algebra' 'Matrix' 'Matrices' 'Inverse')
    6 ## Tagged by tda2d
    7 
    8 ## DBsubject('Linear Algebra')
    9 ## DBchapter('Matrices')
   10 ## DBsection('The Inverse of a Matrix')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('ASU')
   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 "PGasu.pl",
   24 "PG.pl",
   25 "PGbasicmacros.pl",
   26 "PGchoicemacros.pl",
   27 "PGanswermacros.pl",
   28 "PGauxiliaryFunctions.pl",
   29 "PGmatrixmacros.pl"
   30 );
   31 
   32 TEXT(beginproblem());
   33 $showPartialCorrectAnswers = 1;
   34 
   35 $a = non_zero_random(-6,6,1);
   36 $b = non_zero_random(-5,5,1);
   37 $c = non_zero_random(-4,4,1);
   38 do {$d = random(-9,9,2);} until (($a * $d - $b * $c) != 0);
   39 
   40 $B11 = non_zero_random(-3,3,1);
   41 $B21 = non_zero_random(-3,3,1);
   42 
   43 do {$B12 = non_zero_random(-3,3,1);} until ($B12 != $B11);
   44 $B22 = non_zero_random(-4,4,1);
   45 
   46 $det = $a * $d - $b * $c;
   47 
   48 $ans11 = $d / $det;
   49 $ans12 = - $b / $det;
   50 $ans21 = - $c / $det;
   51 $ans22 = $a / $det ;
   52 
   53 $x1 = $ans11*$B11+$ans12*$B21;
   54 $y1 = $ans21*$B11+$ans22*$B21;
   55 
   56 $x2 = $ans11*$B12+$ans12*$B22;
   57 $y2 = $ans21*$B12+$ans22*$B22;
   58 
   59 $ls1 = nicestring([$a,$b],['x','y']);
   60 $ls2 = nicestring([$c,$d],['x','y']);
   61 
   62 
   63 BEGIN_TEXT
   64 Consider the following two systems.
   65 $BR
   66 (a)
   67 \[ \left\{"\{"\} \begin{array}{ccc}
   68 $ls1 &=& $B11 \\
   69 $ls2 &=& $B21
   70 \end{array} \right. \]
   71 $BR
   72 (b)
   73 \[ \left\{"\{"\} \begin{array}{ccc}
   74 $ls1 &=& $B12 \\
   75 $ls2 &=& $B22
   76 \end{array} \right.\]
   77 $BR
   78 (i) Find the inverse of the (common) coefficient matrix of the two systems.
   79 $BR
   80 $BCENTER
   81 \{ mbox(
   82 '\( A^{-1} = \)',
   83 display_matrix([[ans_rule(10),ans_rule(10)],
   84 [ans_rule(10),ans_rule(10)]],
   85 'align'=>"cc")) \}
   86 $ECENTER
   87 
   88 $BR
   89 (ii) Find the solutions to the two systems by using the inverse, i.e. by evaluating
   90 \( A^{-1} B \) where \( B \) represents the right hand side (i.e.
   91 \(B = \left[ \begin{array}{c} $B11 \\ $B21 \end{array} \right] \) for  system (a) and
   92 \(B = \left[ \begin{array}{c} $B12 \\ $B22 \end{array} \right] \) for  system (b)).
   93 $BR
   94 Solution to  system (a):  \( x  = \) \{ ans_rule(10)\} , \(y\) =\{ ans_rule(10) \}
   95 $BR
   96 Solution to system (b):  \(x  =\) \{ans_rule(10)\} , \(y\) =\{ans_rule(10)\}
   97 
   98 END_TEXT
   99 
  100 ANS(num_cmp($ans11));
  101 ANS(num_cmp($ans12));
  102 ANS(num_cmp($ans21));
  103 ANS(num_cmp($ans22));
  104 ANS(num_cmp($x1));
  105 ANS(num_cmp($y1));
  106 ANS(num_cmp($x2));
  107 ANS(num_cmp($y2));
  108 
  109 
  110 
  111 
  112 ENDDOCUMENT();       # This should be the last executable line in the problem.