My question arises from a discussion on an answer given by Maurizio Monge here.I do not know if there is a known terminology for such matrices. By "sign matrices," I mean square matrices whose entries are in ${-1,+1}$.
For instance,
$\begin{bmatrix}
1 &-1 \\
-1& -1
\end{bmatrix}$ ,
$\begin{bmatrix}
-1&1&1 \\
1&1&-1 \\
-1&-1&-1
\end{bmatrix}$
Clearly, there are $2^{n^2}$ sign matrices of size $n\times n$. So, we start their theory by enumerating them as follows. For a matrix of size $n\times n$ we consider a truth table of $n^2$ arguments and therefore $2^{n^2}$ rows. Each row corresponds to the entries in one matrix$(a_{11},a_{12},\dots,a_{1n},a_{21},a_{22},\dots,a_{nn})$. Let $M_{(n,k)}$ be the $n \times n$ sign matrix corresponding to the $k^th$ row of the truth table.
Question: Does the following matrix product give the zero matrix for sign matrices of even size?
$\prod_{k=1}^{2^{n^2}}M_{(n,k)}$
Thank you. As usual, I will be delighted if you point me to good references on this.
