3 edited math display per requested; edited tags

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.