## DESCRIPTION
## Linear Algebra
## ENDDESCRIPTION

## KEYWORDS('matrix' 'determinant')
## Tagged by tda2d

## DBsubject('Linear Algebra')
## DBchapter('Matrices')
## DBsection('Properties of Determinants')
## Date('')
## Author('edited by Shifrin')
## Institution('TCNJ')
## 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"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$det1 = non_zero_random(-5,5,1);

$k = random(2,9,1)*random(-1,1,2);
$j = random(2,9,1)*random(-1,1,2);

$ans1 = $det1*$k;
$ans2 = $det1*$j;
$ans3 = - $det1;

$str1 = $str1 . "$k";
$str1 = $str1 . "c";
$str2 = $str2 . "$k";
$str2 = $str2 . "f";
$str3 = $str3 . "$k";
$str3 = $str3 . "i";

$str4 = $str4 . "$j";
$str4 = $str4 . "b";
$str5 = $str5 . "$j";
$str5 = $str5 . "e";
$str6 = $str6 . "$j";
$str6 = $str6 . "h";

BEGIN_TEXT

Given \( \left| \begin{matrix} a&b&c\\d&e&f\\g&h&i \end{matrix}\right|=$det1 \), find the following determinants.
$PAR
\( \left| \begin{matrix}a&b&$str1\\ d&e&$str2 \\ g&h&$str3 \end{matrix} \right|= \ \) \{ans_rule(5)\}
$PAR
\(\left| \begin{matrix} a&$str4&c\\ d&$str5&f \\ g&$str6&i\end{matrix}\right| = \ \) \{ans_rule(5) \}
$PAR
\( \left|\begin{matrix} c&b&a \\ f&e&d \\ i&h&g\end{matrix}\right|= \ \) \{ans_rule(5) \}

END_TEXT

ANS(num_cmp($ans1));
ANS(num_cmp($ans2));
ANS(num_cmp($ans3));

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