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What is the minimal positive determinant of a $3\times 3$ matrix with non-repeating entries selected from $\{1,2,...,9\}$, such as the following:

\begin{align} \left [ \begin{matrix} 1&2&3 \\ 4&5&6 \\ 7&8&9 \\ \end{matrix} \right ] \end{align}

Gary
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Kevin Xu
  • 101

1 Answers1

0

Here's a script that lists all matrices with determinant of absolute value 1, giving one representative for each equivalence class relative to row/column permutations. For each of these representatives, the upper-left entry is 1, and the middle entry is the smallest within the bottom right $2 \times 2$ square. Note that this does not account for the fact that transposing one valid solution gives you another, so we are still redundantly counting by a factor of at least 2.

import numpy as np
from itertools import permutations, combinations

n = 3


first_row,first_col = [0,0,1,2],[1,2,0,0]
sec_row,sec_col = [1,2,2],[2,1,2]


num_set = np.arange(2,n**2+1)


diagonal: 1,a,_


sols = []
ct = 0
for a in range(2,2n+1):
    for inner in combinations(range(a+1,n*2+1),3):
        outer = np.setdiff1d(num_set,(a,) + inner)
        for p1 in permutations(outer):
            for p2 in permutations(inner):
                A = np.diag([1,a,0])
                A[first_row,first_col] = p1
                A[sec_row,sec_col] = p2
                if abs(round(np.linalg.det(A),0)) == 1:
                    sols.append(A)

Here are the results

1 7 8   1 9 6   1 7 8   1 9 4   1 4 7   1 6 9   1 4 8   1 6 7   
9 2 3   7 2 4   9 2 3   7 2 5   6 2 3   4 2 5   6 2 3   4 2 5   
6 4 5   8 3 5   4 5 6   8 3 6   9 5 8   7 3 8   7 5 9   8 3 9

1 4 5   1 8 6   1 5 7   1 6 4   1 3 6   1 7 8   1 5 7   1 6 3

8 2 3   4 2 7   6 2 9   5 2 3   7 2 5   3 2 4   6 2 9   5 2 4

6 7 9   5 3 9   4 3 8   7 9 8   8 4 9   6 5 9   3 4 8   7 9 8


1 6 5   1 7 3   1 8 2   1 9 6   1 2 7   1 6 9   1 2 6   1 8 7

7 2 4   6 2 9   9 3 7   8 3 4   6 3 4   2 3 5   8 3 5   2 3 4

3 9 8   5 4 8   6 4 5   2 7 5   9 5 8   7 4 8   7 4 9   6 5 9


1 2 5   1 7 9   1 5 2   1 6 9   1 2 8   1 4 9   1 2 7   1 4 9

7 3 6   2 3 4   6 3 4   5 3 8   4 3 5   2 3 6   4 3 5   2 3 6

9 4 8   5 6 8   9 8 7   2 4 7   9 6 7   8 5 7   9 6 8   7 5 8


1 2 8   1 4 7   1 4 6   1 7 2   1 4 5   1 6 2   1 2 3   1 3 2

4 3 6   2 3 5   7 3 9   4 3 5   6 3 8   4 3 7   6 4 7   8 4 5

7 5 9   8 6 9   2 5 8   6 9 8   2 7 9   5 8 9   8 5 9   6 9 7


1 6 8   1 8 6   1 2 3   1 6 3   1 7 2   1 7 6   1 2 5   1 3 9

2 4 5   3 4 9   7 4 8   7 4 8   6 4 9   2 4 5   3 4 8   2 4 7

3 7 9   2 5 7   6 5 9   2 9 5   3 8 5   3 8 9   9 7 6   5 8 6


1 2 3   1 2 4   1 3 8   1 8 4   1 2 3   1 7 4   1 2 4   1 2 4

8 5 7   3 5 9   2 5 7   2 5 6   7 5 8   2 5 6   3 5 7   3 5 9

4 6 9   8 7 6   4 9 6   3 7 9   4 6 9   3 8 9   6 9 8   6 7 8


1 3 2   1 3 2   1 3 6   1 3 6   1 6 4   1 6 4   1 3 2   1 4 5

6 5 8   6 5 9   2 5 7   2 5 9   3 5 8   3 5 9   4 6 7   3 6 8

4 9 7   4 8 7   4 9 8   4 7 8   2 9 7   2 8 7   5 8 9   2 7 9

For your own printing convenience:

for i in range(8):
    for rows in zip(*sols[8*i:8*(i+1)]):
        for row in rows:
            print(*row,end='   ')
        print()
    print()
Ben Grossmann
  • 225,327