I am given that A is a simple finite-dimensional associative unital algebra over the $\mathbb{C}$, and $M$ is a simple $A$-module. Furthermore, $V_M = \text{Hom}_A(M, A)$ is a right A-module with the action $(f.x)(m) = f(m).x$ for $f \in V_M, x \in A$, and $m \in M$. The task is to find the dimension of $V_M$ over $\mathbb{C}$, in terms of "representation theoretical data about $A$".
The only remotely interesting observation I've made is that, for each $m\in M$, there is an injective right A-module homomorphism from $V_M$ into $A$, given by $f \mapsto f(m)$, where injectivity comes from the fact that the kernel of the map is a two-sided ideal of $A$. I have no idea if this is useful, but it's the only inroad I've been able to make.
Any help at all would be very much appreciated.