Let $S=(s\_{ij})$ be a skew-symmetric integral matrix of order $n$. We only consider the case that $n$ is even. Let $e$ be the all-one vector in $\mathbb{R}^n$. Define the walk matrix $$W(S)=[e,Se,\cdots,S^{n-1}e].$$ (the name "walk matrix" comes from graph theory, where $S$ is the adjacency matrix of an undirected graph. Of course, $S$ is not skew-symmetric in the setting of graphs.)


It is well-known that $\det(S)$ is always a square number. I find that the integer $\sqrt{\det(S)}$ is always a divisor of $\det W(S)$. But I cannot find any references on this relation.


For example, consider $$S=\left(
\begin{array}{cccc}
 0 & 4 & 0 & -3 \\
 -4 & 0 & -2 & -1 \\
 0 & 2 & 0 & 3 \\
 3 & 1 & -3 & 0 \\
\end{array}
\right).$$ Then, $$W(S)=\left(
\begin{array}{cccc}
 1 & 1 & -31 & -3 \\
 1 & -7 & -15 & 165 \\
 1 & 5 & -11 & -87 \\
 1 & 1 & -19 & -75 \\
\end{array}
\right).$$
Using Mathematica, we find that $\det(S)=18^2$, $\det(W)=16128=18\times 896$ and $\sqrt{\det(S)}\mid \det(W)$.


It seems that the above relation $\sqrt{\det(S)}\mid \det(W)$ always hold for any skew-symmetric integral matrix of even orders. In particular, when $\det(S)=0$ then $\det(W)=0$. This particular case is not hard to show.



Determinant of walk matrix for a skew-symmetric matrix of even order
Surely, there is nothing special in the all-ones vector: the claim holds for any integer-valued $e$.


Notice that
$$
 \det W^TW
 =\det\bigl[e^T (-1)^iS^{i+j}e\bigr],
$$
Since $S$ is skew-symmetric, we have $e^TS^ie=0$ for all odd $i$. Permuting now the rows and columns of $W^TW$ we get
$$
 \det W^TW=\det\begin{bmatrix} A\_0&0\\ 0& -A\_1\end{bmatrix},
$$
where
$$
 A\_0= \begin{bmatrix} 
 e^Te& e^TS^2e& e^TS^4e& \cdots& e^TS^{n-2}e\\
 e^TS^2e& e^TS^4e& e^TS^6e& \cdots& e^TS^ne\\
 \vdots& \vdots& \vdots& \ddots& \vdots\\
 e^TS^{n-2}e& e^TS^ne& e^TS^{n+2}e& \cdots& e^TS^{2n-4}e
\end{bmatrix}, \quad
 A\_1= \begin{bmatrix} 
 e^TS^2e& e^TS^4e& e^TS^6e& \cdots& e^TS^ne\\
 e^TS^4e& e^TS^6e& e^TS^8e& \cdots& e^TS^{n+2}e\\
 \vdots& \vdots& \vdots& \ddots& \vdots\\
 e^TS^{n}e& e^TS^{n+2}e& e^TS^{n+4}e& \cdots& e^TS^{2n-2}e
\end{bmatrix}.
$$
We show that
$$
 \det A\_1=\pm\det S\det A\_0, \qquad(\*)
$$
so that $\det W=\pm\det A\_0\sqrt{\det S}$, which yields what we want.


Denote
$$
 u\_i= \begin{bmatrix} 
 e^TS^ie& e^TS^{i+2}e& \cdots& e^TS^{i+n-2}e
 \end{bmatrix}.
$$
The eigenvalues of $S$ are purely imaginary, so by Cayley—Hamilton we get
$$
 S^{n}=\sum\_{j=0}^{n/2-1} \alpha\_jS^{2j},
$$
where $\alpha\_0=-\det S$.
Therefore,
$$
 u\_n= -\det S\, u\_0+\sum\_{j=1}^{n/2-1} \alpha\_ju\_{2j}.
$$
Plug this into $\det A\_1$, expand by linearity, and erase vanishing summands to get $(\*)$.


