How many solutions are there to a matrix over a finite field? Given a matrix of the form Ax=b of arbitrary dimensions, which is defined over a finite field Z with p elements (p being a prime number), how many solutions can the matrix have?

A matrix which is defined over R (all reals, infinite field) may have infinite solutions. This is impossible with a finite field.
(EDITED) 
Let $A$ be an $m \times n$ matrix over your field $\mathbb F$.  In the case $A = 0$, $b=0$, all members of $\mathbb F^n$ are solutions, and there are $p^n$ of them.  If $A$ has rank $r$, the solutions (if any) form an affine subspace of dimension $n-r$, so there are $p^{n-r}$ solutions if there are any.
Apparently, such a matrix can have p to the power of n solutions, where n is the amount of free variables in the matrix.