Maximum Determinants

I fell to thinking as to what is the maximum determinant that can arise when the digits from 1 to 9 are entered into a 3 x 3 matrix. I came across this site that stated that the maximum determinant is 412. An example of a matrix with this maximum determinant is:

$$\text{det }\begin{bmatrix} 1 & 4 & 8 \\ 7 & 2 & 6 \\ 5 & 9 & 3 \end{bmatrix} = 412$$ It turns out that there is an OEIS sequence that lists the maximum determinants for matrices up to 7 x 7. It is OEIS A085000 :

  A085000  Maximal determinant of an $n \times n$ matrix using the integers $1$ to $n^2$.

The sequence begins: 1, 10, 412, 40800, 6839492, 1865999570, 762150368499

The 2 x 2 configurations are easy enough as there are only a total 24 possible configurations. One example of a 2 x 2 matrix with the maximum determinant of 10 is:

$$\text{det }\begin{bmatrix} 3 & 1 \\ 2 &4 \end{bmatrix} =10$$ Before moving on to the higher 4 x 4, 5 x 5 etc. determinants, let's look at some maximum determinants where more constraints are imposed on a 3 x 3 matrix than simply entering the digits 1 to 9. For example, let's say we want the determinant to be a palindrome or a square or a cube or a prime. Here are examples of matrices that successively satisfy these additional criteria (source): $$\begin{align} \text{det } \begin{bmatrix} 1 & 4 & 7 \\ 9 & 2 & 6 \\ 5 & 8 & 3 \end{bmatrix} &= 404 \\ \\ \text{det } \begin{bmatrix} 1 & 4 & 8 \\ 7 & 3 & 6 \\ 5 & 9 & 2 \end{bmatrix} &= 400 =20^2 \\ \\ \text{det } \begin{bmatrix} 1 & 4 & 6 \\ 8 & 3 & 5 \\ 7 & 9 & 2 \end{bmatrix} &= 343 = 7^3 \\ \\ \text{det } \begin{bmatrix} 1 & 4 & 9 \\ 8 & 3 & 6 \\ 5 & 7 & 2 \end{bmatrix} &= 389 \end{align}$$ For 3 x 3 magic squares, the determinant is always 360. Here is an example of such a magic square matrix: $$\text{det } \begin{bmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{bmatrix} = 360$$ As can be seen from the OEIS information, the maximum determinant of a 4 x 4 matrix is 40800. Here is an example of such a matrix: $$\text{det } \begin{bmatrix} 16 & 6 & 4 & 9 \\  8 & 13 & 11 & 1 \\ 3 & 12 & 5 & 14 \\ 7 & 2 & 15 & 10 \end{bmatrix}= 40800$$ For a 5 x 5 matrix, the maximum determinant is 6839492. Here is an example of such a matrix (source): $$\text{det } \begin{bmatrix} 25 & 15 & 9 & 11 & 4 \\ 7 & 24 & 14 & 3 & 17 \\ 6 & 12 & 23 & 20 & 5 \\ 10 & 13 & 2 & 22 & 19 \\ 16 & 1 & 18 & 8 & 21 \end{bmatrix} =6839492$$ I won't display examples of the 6 x 6 and 7 x 7 matrices where the maximums are 1865999570 and 762150368499 respectively but I'll instead quote from the OEIS comments:

a(6) found with FORTRAN program given at Pfoertner link. A corresponding matrix is ((36 24 21 17 5 8) ( 3 35 25 15 23 11) (13 7 34 16 10 31) (14 22 2 33 12 28) (20 4 19 29 32 6) (26 18 9 1 30 27) ). - Hugo Pfoertner, Sep 23 2003

a(7) is the determinant of the matrix ((46 42 15 2 27 24 18) (9 48 36 30 7 14 31) (39 11 44 34 13 29 5) (26 22 17 41 47 1 21) (20 8 40 6 33 23 45) (4 28 19 25 38 49 12) (32 16 3 37 10 35 43)). Although no proof for the optimality of a(7) is available, the results of an extensive computational search make the existence of a better solution extremely unlikely. A total of approximately 15 CPU years on SGI Origin 3000 and of 3.8 CPU years on SGI Altix 3000 computers was used for this result.