How many solutions to a matrix/vector multiplication?

Question

Suppose I have a square matrix $A$ (e.g., $n\times n$) that multiplies some vector $y$ ($1\times n$) into a vector of the same arrangement $z$ ($1\times n$), such that $Ay = z$, where both $y$ and $z$ are known but not $A$. How many matrices can solve the equation, and how can I calculate that number?