Given an equation (or a set of equations) involving matrices, is there an algorithm to find possible representations of these matrices?
For example, we can consider a matrix $A$ such that $A^2=\begin{bmatrix}0&0\\0&0\end{bmatrix}$. Now one possible representation could be $A=\begin{bmatrix}0&1\\0&0\end{bmatrix}$, another could be $A=\begin{bmatrix}0&0\\1&0\end{bmatrix}$.
As another example, we can consider an algebra involving two matrices $A$ and $B$ such that $AB=A + B$. Now is it possible to find possible representations of $A$ and $B$? For this particular case, it is known that no finite dimensional representations exist.
But we could consider more general cases involving a larger number of matrices and more equations that they have to satisfy. In that case, are there any general algorithms to find possible representations?