Imagine that I have a "secret" vector $\mathbf{x} = (x_1, \dots, x_n) \in \{0,1\}^n$ and I give another person the matrix $$\mathbf{A_x} := \sum_{i=1}^n x_i \cdot \mathbf{A}_i,$$ with the elements of $\mathbf{A}_i \in\mathbb{Z}_q^{n \times n}$ taken uniformly at random from $\mathbb{Z_q}$, for a prime $q$.
Is there any way I can obtain some information about $\mathbf{x}$ (i.e., knowing the number of $0$'s) by just knowing $\mathbf{A_x}$? Is this a known problem? What would happen if we also provide the other person with the matrices $\mathbf{A}_i$, for $i=1,\dots,n$?
Could it happen that given two distinct secret vectors $\mathbf{x}, \mathbf{y} \in \{0,1\}^n$, we have that $\mathbf{A_x} = \mathbf{A_y}$? Is this probability negligible?