Something a Little Different 2

Algebra Level 3

Let $M$ be the set of all $3 \times 3$ matrices with entries all equal to either $0$ or $-1$ . What is the average determinant of the matrices in $M$ ?

3 -1 1 0 9

1 solution

Kyle T
Jun 24, 2019

<?php
//start with array 0 and -1 (think of this as the number in the top left corner)
$arr = array(array(0), array(-1));
//continue adding numbers until each aray has 9 values (imagine this as a 3x3 matrix reading left to right, top to bottom)
do {
    $newarr = array();
    foreach($arr as $a){
        $newarr[] = array_merge($a,[0]);
        $newarr[] = array_merge($a,[-1]);
    }
    $arr = $newarr;
} while(count($arr[0])<9);

/*
Matrix
$a[0] $a[1] $a[2]
$a[3] $a[4] $a[5]
$a[6] $a[7] $a[8]
*/

//compute the determinant for each array
//https://www.mathsisfun.com/algebra/matrix-determinant.html
$d = array();
foreach($arr as $a){
    $d[] = $a[0]*(($a[4]*$a[8])-($a[5]*$a[7])) - $a[1]*(($a[3]*$a[8])-($a[5]*$a[6])) + $a[2]*(($a[3]*$a[7])-($a[4]*$a[6]));
}

//calculate the average (sum/count), this is our answer
echo array_sum($d)/count($d); //answer: 0

//to better explain why this is our answer, lets look at the different values for our determinants and the count for each value
$dcount = array();
foreach($d as $v){
    if(!isset($dcount[$v])){
        $dcount[$v] = 0;
    }
    $dcount[$v]++;
}
echo '<pre>'.print_r($dcount,true).'</pre>';
/*
Array
(
    [0] => 338
    [1] => 84
    [-1] => 84
    [-2] => 3
    [2] => 3
)
*/

//So we can see most often the determinant is 0, but the values range from -2 to 2. 
//The counts for 1 and -1, as well as 2 and -2 match. 
//So when we take the sum of the array it will return 0; and 0 over anything is still 0.
?>

Now, can you show it mathematically?

Hamza A - 1 year, 11 months ago

Something a Little Different 2

1 solution

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