My question arises from a discussion on an answer given by Maurizio Monge [here](https://mathoverflow.net/questions/40005/generalizing-a-problem-to-make-it-easier/40030#40030).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.



'Sign matrices'-(-1,+1) square matrices
Much is known about sign nonsingular patterns (sign patterns for which nonsingularity does not depend on the numerical values), if I remember correctly there is a characterization. Less is known about sign patterns which have allow (but do not require) nonsingularity. I suggest looking at the book *Matrices of sign-solvable linear systems* by Brualdi and Shader.


