## DESCRIPTION
## Linear Algebra
## ENDDESCRIPTION

## KEYWORDS ('linear algebra','matrix','skew-symmetric')
## Tagged by cmd6a 5/3/06

## DBsubject('Linear Algebra')
## DBchapter('Matrices')
## DBsection('Diagonalization')
## 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 = NchooseK(6,6);
$i1 = random(1,2,1);
$j1 = 3 - $i1;
$i2 = random(1,3,2);
$j2 = 4 - $i2;
$i3 = random(2,3,1);
$j3 = 5 - $i3;
$a[1] = non_zero_random(-5,5,1);
$a[2] = $a[1]+non_zero_random(-3,3,2);
$a[3] = $a[1]+non_zero_random(-4,4,2);


BEGIN_TEXT

Enter a \( 3 \times 3 \) skew-symmetric matrix \(A\) that has entries 
$BR \(a_{$i1 $j1}=$a[1]\), \(a_{$i2 $j2}=$a[2]\), \(a_{$i3 $j3}=$a[3]\). 
$BR
\{ mbox( '\(A=\)', answer_matrix(3,3,5), '.' ) \} 

END_TEXT

if ($i1==1) { 
	$a[12] = $a[1];
	$a[21] = - $a[1];
} else { 
        $a[12] = - $a[1];
        $a[21] = $a[1];
}

if ($i2==1) { 
        $a[13] = $a[2];
        $a[31] = - $a[2];
} else { 
        $a[13] = - $a[2];
        $a[31] = $a[2];
}

if ($i3==2) { 
        $a[23] = $a[3];
        $a[32] = - $a[3];
} else { 
        $a[23] = - $a[3];
        $a[32] = $a[3];
}

ANS(num_cmp(0));  ANS(num_cmp($a[12]));  ANS(num_cmp($a[13]));
ANS(num_cmp($a[21]));  ANS(num_cmp(0));  ANS(num_cmp($a[23]));
ANS(num_cmp($a[31]));  ANS(num_cmp($a[32]));  ANS(num_cmp(0));

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