Minimum and maximum value of determinant of $3 \times 3$ matrices

Question

What are the minimum and maximum values of the determinant of a $3 \times 3$ matrix containing distinct elements from $1$ to $9$?

Answer 1

Using Python:

>>> from itertools import *
>>> from sympy import *
>>> dets = [ Matrix(3,3,p).det() for p in list(permutations(range(1,10))) ]

Finding the minimal and maximal determinants:

>>> ( min(dets), max(dets) )
(-412, 412)

These values are in agreement with Ray's answer.

Answer 2

Brute Force Calculation in Java: I found that the max is 412 and the minimum is -412

package mathstesting;

/**
 *
 * @author Ray Lin
 */
public class MathsTesting
{
    static int max = -1000;
    static int min = 1000; 

    public static void main(String[] args)
    { 
        for(int a = 1; a<=9; a++) //<=9
        {
            for(int b = 1; b<=9; b++)
            {
                if(b != a)
                {
                    for(int c = 1; c<=9; c++)
                    {
                        if((c!= a) && (c!=b))
                        {
                            for(int d = 1; d<=9; d++)
                            {
                                if((d!= a) && (d!=b) && (d!=c))
                                {
                                    for(int e = 1; e<=9; e++)
                                    {
                                        if((e!= a) && (e!=b) && (e!=c) && (e!=d))
                                        {
                                            for(int f = 1; f<=9; f++)
                                            {
                                                if((f!= a) && (f!=b) && (f!=c) && (f!=d) && (f!=e))
                                                {   
                                                    for(int g = 1; g<=9; g++)
                                                    {
                                                        if((g!= a) && (g!=b) && (g!=c) && (g!=d) && (g!=e) && (g!=f))
                                                        {
                                                            for(int h = 1; h<=9; h++)
                                                            {
                                                                if((h!= a) && (h!=b) && (h!=c) && (h!=d) && (h!=e) && (h!=f) && (h!=g))
                                                                {
                                                                    for(int i = 1; i<=9; i++)
                                                                    {
                                                                        if((i!= a) && (i!=b) && (i!=c) && (i!=d) && (i!=e) && (i!=f) && (i!=g)&&(i!=h))
                                                                        {
                                                                            det(a,b,c,d,e,f,g,h,i);
                                                                        }
                                                                    }
                                                                }
                                                            }
                                                        }
                                                    }
                                                }
                                            }
                                        }
                                    }
                                }
                            }
                        }
                    }
                }
            }

        }
    }

    public static void det(int a,int b,int c,int d, int e,int f, int g, int h, int i)
    {
        determinant(a,b,c,d,e,f,g,h,i);
    }

    public static int determinant(int a,int b,int c,int d, int e,int f, int g, int h, int i)
    {
        int temp =  ((a*e*i)+(b*f*g)+(c*d*h)-(a*f*h)-(b*d*i)-(c*e*g));
        if(temp >= max)
        {
            System.out.println("Maximum");
            System.out.println(a + "," + b + ","+ c + ","+ d + "," + e + "," + f+ ","+g+ ","+h+ ","+i+ ",");
            System.out.print( " det is: " + temp);
            System.out.println();
        }
        if(temp <= min)
        {
            System.out.println("Minimum");
            System.out.println(a + "," + b + ","+ c + ","+ d + "," + e + "," + f+ ","+g+ ","+h+ ","+i+ ",");
            System.out.print( " det is: " + temp);
            System.out.println();
        }
        return temp;
    }
}

results are:

Minimum is -412, Maximum is 412

Solutions: Minimum 1,4,8,5,9,3,7,2,6, det is: -412 Maximum 1,4,8,7,2,6,5,9,3, det is: 412 Minimum 1,5,7,4,9,2,8,3,6, det is: -412 Maximum 1,5,7,8,3,6,4,9,2, det is: 412 Maximum 1,7,5,4,2,9,8,6,3, det is: 412 Minimum 1,7,5,8,6,3,4,2,9, det is: -412 Maximum 1,8,4,5,3,9,7,6,2, det is: 412 Minimum 1,8,4,7,6,2,5,3,9, det is: -412 Minimum 2,4,9,6,8,3,7,1,5, det is: -412 Maximum 2,4,9,7,1,5,6,8,3, det is: 412 Minimum 2,6,7,4,8,1,9,3,5, det is: -412 Maximum 2,6,7,9,3,5,4,8,1, det is: 412 Maximum 2,7,6,4,1,8,9,5,3, det is: 412 Minimum 2,7,6,9,5,3,4,1,8, det is: -412 Maximum 2,9,4,6,3,8,7,5,1, det is: 412 Minimum 2,9,4,7,5,1,6,3,8, det is: -412 Minimum 3,5,9,6,7,2,8,1,4, det is: -412 Maximum 3,5,9,8,1,4,6,7,2, det is: 412 Minimum 3,6,8,5,7,1,9,2,4, det is: -412 Maximum 3,6,8,9,2,4,5,7,1, det is: 412 Maximum 3,8,6,5,1,7,9,4,2, det is: 412 Minimum 3,8,6,9,4,2,5,1,7, det is: -412 Maximum 3,9,5,6,2,7,8,4,1, det is: 412 Minimum 3,9,5,8,4,1,6,2,7, det is: -412 Minimum 4,1,8,2,7,6,9,5,3, det is: -412 Maximum 4,1,8,9,5,3,2,7,6, det is: 412 Minimum 4,2,9,1,7,5,8,6,3, det is: -412 Maximum 4,2,9,8,6,3,1,7,5, det is: 412 Maximum 4,8,1,2,6,7,9,3,5, det is: 412 Minimum 4,8,1,9,3,5,2,6,7, det is: -412 Maximum 4,9,2,1,5,7,8,3,6, det is: 412 Minimum 4,9,2,8,3,6,1,5,7, det is: -412 Minimum 5,1,7,3,8,6,9,4,2, det is: -412 Maximum 5,1,7,9,4,2,3,8,6, det is: 412 Minimum 5,3,9,1,8,4,7,6,2, det is: -412 Maximum 5,3,9,7,6,2,1,8,4, det is: 412 Maximum 5,7,1,3,6,8,9,2,4, det is: 412 Minimum 5,7,1,9,2,4,3,6,8, det is: -412 Maximum 5,9,3,1,4,8,7,2,6, det is: 412 Minimum 5,9,3,7,2,6,1,4,8, det is: -412 Minimum 6,2,7,3,9,5,8,4,1, det is: -412 Maximum 6,2,7,8,4,1,3,9,5, det is: 412 Minimum 6,3,8,2,9,4,7,5,1, det is: -412 Maximum 6,3,8,7,5,1,2,9,4, det is: 412 Maximum 6,7,2,3,5,9,8,1,4, det is: 412 Minimum 6,7,2,8,1,4,3,5,9, det is: -412 Maximum 6,8,3,2,4,9,7,1,5, det is: 412 Minimum 6,8,3,7,1,5,2,4,9, det is: -412 Minimum 7,1,5,2,4,9,6,8,3, det is: -412 Maximum 7,1,5,6,8,3,2,4,9, det is: 412 Minimum 7,2,6,1,4,8,5,9,3, det is: -412 Maximum 7,2,6,5,9,3,1,4,8, det is: 412 Maximum 7,5,1,2,9,4,6,3,8, det is: 412 Minimum 7,5,1,6,3,8,2,9,4, det is: -412 Maximum 7,6,2,1,8,4,5,3,9, det is: 412 Minimum 7,6,2,5,3,9,1,8,4, det is: -412 Minimum 8,1,4,3,5,9,6,7,2, det is: -412 Maximum 8,1,4,6,7,2,3,5,9, det is: 412 Minimum 8,3,6,1,5,7,4,9,2, det is: -412 Maximum 8,3,6,4,9,2,1,5,7, det is: 412 Maximum 8,4,1,3,9,5,6,2,7, det is: 412 Minimum 8,4,1,6,2,7,3,9,5, det is: -412 Maximum 8,6,3,1,7,5,4,2,9, det is: 412 Minimum 8,6,3,4,2,9,1,7,5, det is: -412 Minimum 9,2,4,3,6,8,5,7,1, det is: -412 Maximum 9,2,4,5,7,1,3,6,8, det is: 412 Minimum 9,3,5,2,6,7,4,8,1, det is: -412 Maximum 9,3,5,4,8,1,2,6,7, det is: 412 Maximum 9,4,2,3,8,6,5,1,7, det is: 412 Minimum 9,4,2,5,1,7,3,8,6, det is: -412 Maximum 9,5,3,2,7,6,4,1,8, det is: 412 Minimum 9,5,3,4,1,8,2,7,6, det is: -412 BUILD SUCCESSFUL (total time: 0 seconds)