'Sign matrices'-(-1,+1) square matrices 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.
 A: 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.
A: It is not part of the question but I consider it useful to give the result for the $M_3$ case( which I found using a $C$++ program). There are 512 rows and the matrix below is $M_3$. However, I am not completely certain since the compiler I used might have rounded off the numbers while multiplying the matrices though the data type used was long double,the most precise by far.
\begin{matrix}
-1 & 1 & -1 & 1 & -1 & -1 & -1 & -1 & 1\\  
-1 & 1 & -1 & 1 & -1 & -1 & -1 & -1 & -1\\  
-1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & 1\\  
-1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1\\  
-1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & 1\\  
-1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1\\  
\end{matrix}
$$\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$$
\begin{matrix}
-1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & -1\\  
-1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 1\\  
-1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1  
\end{matrix}
\begin{bmatrix}
1.50197\times10^{100} & 1.50197\times10^{100} & 1.50197\times10^{100}\\ 
1.50197\times10^{100} & 1.50197\times10^{100} &1.50197\times10^{100} \\ 
1.50197\times10^{100}& 1.50197\times10^{100} &1.50197\times10^{100} 
\end{bmatrix}
It is sad that the "array size" blows out of the compiler's capacity when running my program for the $M_4$ case, the case that would either kill or save my question.
