Ways to fill up a $4 \times 4$ matrix with numbers $1$ to $16$, s.t its rank is $1$ under modulo $17$

Question

As the title suggest. How many ways are there to fill up a $4 \times 4$ matrix with numbers $1$ to $16$, so that its rank is $1$ under modulo $17$ (i.e. consider elements of the matrix under $Z_{17}$)

Rank being 1 basically means all rows or columns are linearly dependent. If we look at a simple case, $2 \times 2$ matrix with numbers $1$ to $4$, under modulo $5$, all I can find is

\begin{matrix} 1 & 4\\ 3 & 2 \end{matrix}

and its rotations. But I am not sure how to generalize this to higher orders..

Answer 1

I wrote a python script to find them by brute force. The matrices all have $1$ in the first row and first column, and the entries in both the first row and first column are in increasing order (viewing them temporarily as elements of $\mathbb{Z}$.) I found $256$ such matrices. Of course, the rows and columns can each be permuted without changing the rank, so each matrix represents $24^2$ matrices.

I don't see any patterns at all. $256=16^2$ gives one hope, but for me, at least, it's an empty promise. Because of that, I'm appending a list of all $256$ matrices. Maybe you'll see something I don't.

In the listing below, the matrices have been flattened for typographical considerations, so the first line represents the matrix, $$\begin{bmatrix} 1 &2 &3 &6\\ 4 &8 &12 &7\\ 13 &9 &5 &10\\ 16 &15 &14 &11\end{bmatrix}$$

[ 1  2  3  6  4  8 12  7 13  9  5 10 16 15 14 11]
[ 1  2  3  7  4  8 12 11 13  9  5  6 16 15 14 10]
[ 1  2  3 10  4  8 12  6 13  9  5 11 16 15 14  7]
[ 1  2  3 11  4  8 12 10 13  9  5  7 16 15 14  6]
[ 1  2  4  8  3  6 12  7 14 11  5 10 16 15 13  9]
[ 1  2  4  8  5 10  3  6 12  7 14 11 16 15 13  9]
[ 1  2  4  8  6 12  7 14 11  5 10  3 16 15 13  9]
[ 1  2  4  8  7 14 11  5 10  3  6 12 16 15 13  9]
[ 1  2  4  9  3  6 12 10 14 11  5  7 16 15 13  8]
[ 1  2  4  9  5 10  3 11 12  7 14  6 16 15 13  8]
[ 1  2  4  9  6 12  7  3 11  5 10 14 16 15 13  8]
[ 1  2  4  9  7 14 11 12 10  3  6  5 16 15 13  8]
[ 1  2  5  6  4  8  3  7 13  9 14 10 16 15 12 11]
[ 1  2  5  7  4  8  3 11 13  9 14  6 16 15 12 10]
[ 1  2  5 10  4  8  3  6 13  9 14 11 16 15 12  7]
[ 1  2  5 11  4  8  3 10 13  9 14  7 16 15 12  6]

[ 1  2  6 12  4  8  7 14 13  9 10  3 16 15 11  5]
[ 1  2  6 14  4  8  7  5 13  9 10 12 16 15 11  3]
[ 1  2  7 12  4  8 11 14 13  9  6  3 16 15 10  5]
[ 1  2  7 14  4  8 11  5 13  9  6 12 16 15 10  3]
[ 1  2  8 13  3  6  7  5 14 11 10 12 16 15  9  4]
[ 1  2  8 13  5 10  6 14 12  7 11  3 16 15  9  4]
[ 1  2  8 13  6 12 14 10 11  5  3  7 16 15  9  4]
[ 1  2  8 13  7 14  5  6 10  3 12 11 16 15  9  4]
[ 1  2  9 13  3  6 10  5 14 11  7 12 16 15  8  4]
[ 1  2  9 13  5 10 11 14 12  7  6  3 16 15  8  4]
[ 1  2  9 13  6 12  3 10 11  5 14  7 16 15  8  4]
[ 1  2  9 13  7 14 12  6 10  3  5 11 16 15  8  4]
[ 1  2 10 12  4  8  6 14 13  9 11  3 16 15  7  5]
[ 1  2 10 14  4  8  6  5 13  9 11 12 16 15  7  3]
[ 1  2 11 12  4  8 10 14 13  9  7  3 16 15  6  5]
[ 1  2 11 14  4  8 10  5 13  9  7 12 16 15  6  3]

[ 1  2 15 16  3  6 11 14  4  8  9 13  5 10  7 12]
[ 1  2 15 16  3  6 11 14  4  8  9 13 12  7 10  5]
[ 1  2 15 16  3  6 11 14  5 10  7 12 13  9  8  4]
[ 1  2 15 16  3  6 11 14 12  7 10  5 13  9  8  4]
[ 1  2 15 16  4  8  9 13  5 10  7 12 14 11  6  3]
[ 1  2 15 16  4  8  9 13  6 12  5 11  7 14  3 10]
[ 1  2 15 16  4  8  9 13  6 12  5 11 10  3 14  7]
[ 1  2 15 16  4  8  9 13  7 14  3 10 11  5 12  6]
[ 1  2 15 16  4  8  9 13 10  3 14  7 11  5 12  6]
[ 1  2 15 16  4  8  9 13 12  7 10  5 14 11  6  3]
[ 1  2 15 16  5 10  7 12 13  9  8  4 14 11  6  3]
[ 1  2 15 16  6 12  5 11  7 14  3 10 13  9  8  4]
[ 1  2 15 16  6 12  5 11 10  3 14  7 13  9  8  4]
[ 1  2 15 16  7 14  3 10 11  5 12  6 13  9  8  4]
[ 1  2 15 16 10  3 14  7 11  5 12  6 13  9  8  4]
[ 1  2 15 16 12  7 10  5 13  9  8  4 14 11  6  3]

[ 1  3  4  5  2  6  8 10 15 11  9  7 16 14 13 12]
[ 1  3  4  5  8  7 15  6  9 10  2 11 16 14 13 12]
[ 1  3  4 12  2  6  8  7 15 11  9 10 16 14 13  5]
[ 1  3  4 12  8  7 15 11  9 10  2  6 16 14 13  5]
[ 1  3  5 13  2  6 10  9 15 11  7  8 16 14 12  4]
[ 1  3  5 13  8  7  6  2  9 10 11 15 16 14 12  4]
[ 1  3  6  8  4 12  7 15 13  5 10  2 16 14 11  9]
[ 1  3  6  9  4 12  7  2 13  5 10 15 16 14 11  8]
[ 1  3  6 15  4 12  7  9 13  5 10  8 16 14 11  2]
[ 1  3  7  8  4 12 11 15 13  5  6  2 16 14 10  9]
[ 1  3  7  9  4 12 11  2 13  5  6 15 16 14 10  8]
[ 1  3  7 15  4 12 11  9 13  5  6  8 16 14 10  2]
[ 1  3  8 10  4 12 15  6 13  5  2 11 16 14  9  7]
[ 1  3  8 11  4 12 15 10 13  5  2  7 16 14  9  6]
[ 1  3  9 10  4 12  2  6 13  5 15 11 16 14  8  7]
[ 1  3  9 11  4 12  2 10 13  5 15  7 16 14  8  6]

[ 1  3 10 15  4 12  6  9 13  5 11  8 16 14  7  2]
[ 1  3 11 15  4 12 10  9 13  5  7  8 16 14  6  2]
[ 1  3 12 13  2  6  7  9 15 11 10  8 16 14  5  4]
[ 1  3 12 13  8  7 11  2  9 10  6 15 16 14  5  4]
[ 1  3 14 16  2  6 11 15  4 12  5 13  8  7 10  9]
[ 1  3 14 16  2  6 11 15  4 12  5 13  9 10  7  8]
[ 1  3 14 16  2  6 11 15  8  7 10  9 13  5 12  4]
[ 1  3 14 16  2  6 11 15  9 10  7  8 13  5 12  4]
[ 1  3 14 16  4 12  5 13  8  7 10  9 15 11  6  2]
[ 1  3 14 16  4 12  5 13  9 10  7  8 15 11  6  2]
[ 1  3 14 16  8  7 10  9 13  5 12  4 15 11  6  2]
[ 1  3 14 16  9 10  7  8 13  5 12  4 15 11  6  2]
[ 1  4  5 14  2  8 10 11 15  9  7  6 16 13 12  3]
[ 1  4  5 14  8 15  6 10  9  2 11  7 16 13 12  3]
[ 1  4  6  7  2  8 12 14 15  9  5  3 16 13 11 10]
[ 1  4  6  7  8 15 14  5  9  2  3 12 16 13 11 10]

[ 1  4  6 10  2  8 12  3 15  9  5 14 16 13 11  7]
[ 1  4  6 10  8 15 14 12  9  2  3  5 16 13 11  7]
[ 1  4  7 11  2  8 14  5 15  9  3 12 16 13 10  6]
[ 1  4  7 11  8 15  5  3  9  2 12 14 16 13 10  6]
[ 1  4  8 15  3 12  7 11 14  5 10  6 16 13  9  2]
[ 1  4  8 15  5  3  6  7 12 14 11 10 16 13  9  2]
[ 1  4  8 15  6  7 14  5 11 10  3 12 16 13  9  2]
[ 1  4  8 15  7 11  5  3 10  6 12 14 16 13  9  2]
[ 1  4  9 15  3 12 10 11 14  5  7  6 16 13  8  2]
[ 1  4  9 15  5  3 11  7 12 14  6 10 16 13  8  2]
[ 1  4  9 15  6  7  3  5 11 10 14 12 16 13  8  2]
[ 1  4  9 15  7 11 12  3 10  6  5 14 16 13  8  2]
[ 1  4 10 11  2  8  3  5 15  9 14 12 16 13  7  6]
[ 1  4 10 11  8 15 12  3  9  2  5 14 16 13  7  6]
[ 1  4 12 14  2  8  7 11 15  9 10  6 16 13  5  3]
[ 1  4 12 14  8 15 11 10  9  2  6  7 16 13  5  3]

[ 1  4 13 16  2  8  9 15  3 12  5 14  6  7 10 11]
[ 1  4 13 16  2  8  9 15  3 12  5 14  7 11  6 10]
[ 1  4 13 16  2  8  9 15  3 12  5 14 10  6 11  7]
[ 1  4 13 16  2  8  9 15  3 12  5 14 11 10  7  6]
[ 1  4 13 16  2  8  9 15  5  3 14 12  6  7 10 11]
[ 1  4 13 16  2  8  9 15  5  3 14 12  7 11  6 10]
[ 1  4 13 16  2  8  9 15  5  3 14 12 10  6 11  7]
[ 1  4 13 16  2  8  9 15  5  3 14 12 11 10  7  6]
[ 1  4 13 16  2  8  9 15  6  7 10 11 12 14  3  5]
[ 1  4 13 16  2  8  9 15  6  7 10 11 14  5 12  3]
[ 1  4 13 16  2  8  9 15  7 11  6 10 12 14  3  5]
[ 1  4 13 16  2  8  9 15  7 11  6 10 14  5 12  3]
[ 1  4 13 16  2  8  9 15 10  6 11  7 12 14  3  5]
[ 1  4 13 16  2  8  9 15 10  6 11  7 14  5 12  3]
[ 1  4 13 16  2  8  9 15 11 10  7  6 12 14  3  5]
[ 1  4 13 16  2  8  9 15 11 10  7  6 14  5 12  3]

[ 1  4 13 16  3 12  5 14  6  7 10 11  8 15  2  9]
[ 1  4 13 16  3 12  5 14  6  7 10 11  9  2 15  8]
[ 1  4 13 16  3 12  5 14  6  7 10 11 15  9  8  2]
[ 1  4 13 16  3 12  5 14  7 11  6 10  8 15  2  9]
[ 1  4 13 16  3 12  5 14  7 11  6 10  9  2 15  8]
[ 1  4 13 16  3 12  5 14  7 11  6 10 15  9  8  2]
[ 1  4 13 16  3 12  5 14  8 15  2  9 10  6 11  7]
[ 1  4 13 16  3 12  5 14  8 15  2  9 11 10  7  6]
[ 1  4 13 16  3 12  5 14  9  2 15  8 10  6 11  7]
[ 1  4 13 16  3 12  5 14  9  2 15  8 11 10  7  6]
[ 1  4 13 16  3 12  5 14 10  6 11  7 15  9  8  2]
[ 1  4 13 16  3 12  5 14 11 10  7  6 15  9  8  2]
[ 1  4 13 16  5  3 14 12  6  7 10 11  8 15  2  9]
[ 1  4 13 16  5  3 14 12  6  7 10 11  9  2 15  8]
[ 1  4 13 16  5  3 14 12  6  7 10 11 15  9  8  2]
[ 1  4 13 16  5  3 14 12  7 11  6 10  8 15  2  9]

[ 1  4 13 16  5  3 14 12  7 11  6 10  9  2 15  8]
[ 1  4 13 16  5  3 14 12  7 11  6 10 15  9  8  2]
[ 1  4 13 16  5  3 14 12  8 15  2  9 10  6 11  7]
[ 1  4 13 16  5  3 14 12  8 15  2  9 11 10  7  6]
[ 1  4 13 16  5  3 14 12  9  2 15  8 10  6 11  7]
[ 1  4 13 16  5  3 14 12  9  2 15  8 11 10  7  6]
[ 1  4 13 16  5  3 14 12 10  6 11  7 15  9  8  2]
[ 1  4 13 16  5  3 14 12 11 10  7  6 15  9  8  2]
[ 1  4 13 16  6  7 10 11  8 15  2  9 12 14  3  5]
[ 1  4 13 16  6  7 10 11  8 15  2  9 14  5 12  3]
[ 1  4 13 16  6  7 10 11  9  2 15  8 12 14  3  5]
[ 1  4 13 16  6  7 10 11  9  2 15  8 14  5 12  3]
[ 1  4 13 16  6  7 10 11 12 14  3  5 15  9  8  2]
[ 1  4 13 16  6  7 10 11 14  5 12  3 15  9  8  2]
[ 1  4 13 16  7 11  6 10  8 15  2  9 12 14  3  5]
[ 1  4 13 16  7 11  6 10  8 15  2  9 14  5 12  3]

[ 1  4 13 16  7 11  6 10  9  2 15  8 12 14  3  5]
[ 1  4 13 16  7 11  6 10  9  2 15  8 14  5 12  3]
[ 1  4 13 16  7 11  6 10 12 14  3  5 15  9  8  2]
[ 1  4 13 16  7 11  6 10 14  5 12  3 15  9  8  2]
[ 1  4 13 16  8 15  2  9 10  6 11  7 12 14  3  5]
[ 1  4 13 16  8 15  2  9 10  6 11  7 14  5 12  3]
[ 1  4 13 16  8 15  2  9 11 10  7  6 12 14  3  5]
[ 1  4 13 16  8 15  2  9 11 10  7  6 14  5 12  3]
[ 1  4 13 16  9  2 15  8 10  6 11  7 12 14  3  5]
[ 1  4 13 16  9  2 15  8 10  6 11  7 14  5 12  3]
[ 1  4 13 16  9  2 15  8 11 10  7  6 12 14  3  5]
[ 1  4 13 16  9  2 15  8 11 10  7  6 14  5 12  3]
[ 1  4 13 16 10  6 11  7 12 14  3  5 15  9  8  2]
[ 1  4 13 16 10  6 11  7 14  5 12  3 15  9  8  2]
[ 1  4 13 16 11 10  7  6 12 14  3  5 15  9  8  2]
[ 1  4 13 16 11 10  7  6 14  5 12  3 15  9  8  2]

[ 1  5  6  8  4  3  7 15 13 14 10  2 16 12 11  9]
[ 1  5  6  9  4  3  7  2 13 14 10 15 16 12 11  8]
[ 1  5  6 15  4  3  7  9 13 14 10  8 16 12 11  2]
[ 1  5  7  8  4  3 11 15 13 14  6  2 16 12 10  9]
[ 1  5  7  9  4  3 11  2 13 14  6 15 16 12 10  8]
[ 1  5  7 15  4  3 11  9 13 14  6  8 16 12 10  2]
[ 1  5  8 10  4  3 15  6 13 14  2 11 16 12  9  7]
[ 1  5  8 11  4  3 15 10 13 14  2  7 16 12  9  6]
[ 1  5  9 10  4  3  2  6 13 14 15 11 16 12  8  7]
[ 1  5  9 11  4  3  2 10 13 14 15  7 16 12  8  6]
[ 1  5 10 15  4  3  6  9 13 14 11  8 16 12  7  2]
[ 1  5 11 15  4  3 10  9 13 14  7  8 16 12  6  2]
[ 1  5 12 16  2 10  7 15  4  3 14 13  8  6 11  9]
[ 1  5 12 16  2 10  7 15  4  3 14 13  9 11  6  8]
[ 1  5 12 16  2 10  7 15  8  6 11  9 13 14  3  4]
[ 1  5 12 16  2 10  7 15  9 11  6  8 13 14  3  4]

[ 1  5 12 16  4  3 14 13  8  6 11  9 15  7 10  2]
[ 1  5 12 16  4  3 14 13  9 11  6  8 15  7 10  2]
[ 1  5 12 16  8  6 11  9 13 14  3  4 15  7 10  2]
[ 1  5 12 16  9 11  6  8 13 14  3  4 15  7 10  2]
[ 1  5 13 14  2 10  9 11 15  7  8  6 16 12  4  3]
[ 1  5 13 14  8  6  2 10  9 11 15  7 16 12  4  3]
[ 1  6  7 13  2 12 14  9 15  5  3  8 16 11 10  4]
[ 1  6  7 13  8 14  5  2  9  3 12 15 16 11 10  4]
[ 1  6  8 12  4  7 15 14 13 10  2  3 16 11  9  5]
[ 1  6  8 14  4  7 15  5 13 10  2 12 16 11  9  3]
[ 1  6  9 12  4  7  2 14 13 10 15  3 16 11  8  5]
[ 1  6  9 14  4  7  2  5 13 10 15 12 16 11  8  3]
[ 1  6 10 13  2 12  3  9 15  5 14  8 16 11  7  4]
[ 1  6 10 13  8 14 12  2  9  3  5 15 16 11  7  4]
[ 1  6 11 16  2 12  5 15  4  7 10 13  8 14  3  9]
[ 1  6 11 16  2 12  5 15  4  7 10 13  9  3 14  8]

[ 1  6 11 16  2 12  5 15  8 14  3  9 13 10  7  4]
[ 1  6 11 16  2 12  5 15  9  3 14  8 13 10  7  4]
[ 1  6 11 16  4  7 10 13  8 14  3  9 15  5 12  2]
[ 1  6 11 16  4  7 10 13  9  3 14  8 15  5 12  2]
[ 1  6 11 16  8 14  3  9 13 10  7  4 15  5 12  2]
[ 1  6 11 16  9  3 14  8 13 10  7  4 15  5 12  2]
[ 1  6 12 15  4  7 14  9 13 10  3  8 16 11  5  2]
[ 1  6 14 15  4  7  5  9 13 10 12  8 16 11  3  2]
[ 1  7  8 12  4 11 15 14 13  6  2  3 16 10  9  5]
[ 1  7  8 14  4 11 15  5 13  6  2 12 16 10  9  3]
[ 1  7  9 12  4 11  2 14 13  6 15  3 16 10  8  5]
[ 1  7  9 14  4 11  2  5 13  6 15 12 16 10  8  3]
[ 1  7 10 16  2 14  3 15  4 11  6 13  8  5 12  9]
[ 1  7 10 16  2 14  3 15  4 11  6 13  9 12  5  8]
[ 1  7 10 16  2 14  3 15  8  5 12  9 13  6 11  4]
[ 1  7 10 16  2 14  3 15  9 12  5  8 13  6 11  4]

[ 1  7 10 16  4 11  6 13  8  5 12  9 15  3 14  2]
[ 1  7 10 16  4 11  6 13  9 12  5  8 15  3 14  2]
[ 1  7 10 16  8  5 12  9 13  6 11  4 15  3 14  2]
[ 1  7 10 16  9 12  5  8 13  6 11  4 15  3 14  2]
[ 1  7 11 13  2 14  5  9 15  3 12  8 16 10  6  4]
[ 1  7 11 13  8  5  3  2  9 12 14 15 16 10  6  4]
[ 1  7 12 15  4 11 14  9 13  6  3  8 16 10  5  2]
[ 1  7 14 15  4 11  5  9 13  6 12  8 16 10  3  2]
[ 1  8  9 16  3  7 10 14  4 15  2 13  5  6 11 12]
[ 1  8  9 16  3  7 10 14  4 15  2 13 12 11  6  5]
[ 1  8  9 16  3  7 10 14  5  6 11 12 13  2 15  4]
[ 1  8  9 16  3  7 10 14 12 11  6  5 13  2 15  4]
[ 1  8  9 16  4 15  2 13  5  6 11 12 14 10  7  3]
[ 1  8  9 16  4 15  2 13  6 14  3 11  7  5 12 10]
[ 1  8  9 16  4 15  2 13  6 14  3 11 10 12  5  7]
[ 1  8  9 16  4 15  2 13  7  5 12 10 11  3 14  6]

[ 1  8  9 16  4 15  2 13 10 12  5  7 11  3 14  6]
[ 1  8  9 16  4 15  2 13 12 11  6  5 14 10  7  3]
[ 1  8  9 16  5  6 11 12 13  2 15  4 14 10  7  3]
[ 1  8  9 16  6 14  3 11  7  5 12 10 13  2 15  4]
[ 1  8  9 16  6 14  3 11 10 12  5  7 13  2 15  4]
[ 1  8  9 16  7  5 12 10 11  3 14  6 13  2 15  4]
[ 1  8  9 16 10 12  5  7 11  3 14  6 13  2 15  4]
[ 1  8  9 16 12 11  6  5 13  2 15  4 14 10  7  3]
[ 1  8 10 12  4 15  6 14 13  2 11  3 16  9  7  5]
[ 1  8 10 14  4 15  6  5 13  2 11 12 16  9  7  3]
[ 1  8 11 12  4 15 10 14 13  2  7  3 16  9  6  5]
[ 1  8 11 14  4 15 10  5 13  2  7 12 16  9  6  3]
[ 1  8 13 15  3  7  5 11 14 10 12  6 16  9  4  2]
[ 1  8 13 15  5  6 14  7 12 11  3 10 16  9  4  2]
[ 1  8 13 15  6 14 10  5 11  3  7 12 16  9  4  2]
[ 1  8 13 15  7  5  6  3 10 12 11 14 16  9  4  2]

[ 1  9 10 12  4  2  6 14 13 15 11  3 16  8  7  5]
[ 1  9 10 14  4  2  6  5 13 15 11 12 16  8  7  3]
[ 1  9 11 12  4  2 10 14 13 15  7  3 16  8  6  5]
[ 1  9 11 14  4  2 10  5 13 15  7 12 16  8  6  3]
[ 1  9 13 15  3 10  5 11 14  7 12  6 16  8  4  2]
[ 1  9 13 15  5 11 14  7 12  6  3 10 16  8  4  2]
[ 1  9 13 15  6  3 10  5 11 14  7 12 16  8  4  2]
[ 1  9 13 15  7 12  6  3 10  5 11 14 16  8  4  2]
[ 1 10 11 13  2  3  5  9 15 14 12  8 16  7  6  4]
[ 1 10 11 13  8 12  3  2  9  5 14 15 16  7  6  4]
[ 1 10 12 15  4  6 14  9 13 11  3  8 16  7  5  2]
[ 1 10 14 15  4  6  5  9 13 11 12  8 16  7  3  2]
[ 1 11 12 15  4 10 14  9 13  7  3  8 16  6  5  2]
[ 1 11 14 15  4 10  5  9 13  7 12  8 16  6  3  2]
[ 1 12 13 14  2  7  9 11 15 10  8  6 16  5  4  3]
[ 1 12 13 14  8 11  2 10  9  6 15  7 16  5  4  3]