1- Consider a matrix equation as $Q_{m \times m}X_{m \times n} S_{n \times n} = Z_{m \times n}$ where we know that $Q$ and $S$ are full rank matrices. Then, we know that the solution is $X = Q^{-1}ZS^{-1}$.
2- Now consider the following system of equations:
$$ \sum_{j=1}^N Q_{m \times m}^{ij}X_{m \times n}^j S_{n \times n}^{ji} = Z_{m \times n}^i \quad i=1,2,\ldots, N. $$ where $Q^{ij}$ and $S^{ji}$, $i,j=1,2,\ldots, N$ are full rank matrices.
Does there exist any $X^j$ matrices for $j=1,2,\ldots, N$ satisfying this equation (at least one)? It seems that problem 2 is similar to problem 1, but I don't know how to solve it. Any idea?