One thing you can do for smoke testing is to take a known solution $x$ (you could generate it randomly or pick something convenient) and a known matrix $A$, and then set $b$ equal to $Ax$, and solve the equation $Ax = b$ to see if you recover your known solution. This method is known as the method of manufactured solutions.
To avoid the influence of convergence issues, you probably want to make sure that $A$ is well-conditioned. One way to manufacture a well-conditioned $A$ would be to select a collection of eigenvalues that are in a bounded interval, and construct the diagonal matrix $\Lambda$ such that the main diagonal consists of those eigenvalues. Then, if you generate an invertible matrix $V$ (also should be well conditioned), $A = V \Lambda V^{-1}$. It's probably best to let $V$ be orthogonal in some of your test cases, in which case $V^{-1} = V^{T}$ and you avoid conditioning issues at the cost of making $A$ symmetric.
By no means will such a procedure substitute for using the matrices in collections like the Matrix Market and the University of Florida sparse matrix collection, but by generating small matrices and using the method of manufactured solutions, you should be able to detect and correct simple errors (say, due to indexing, or faulty logic) without having to run large test cases. It's also a methodology that will generalize to other types of problems (testing ODE solvers, testing PDE solvers, and so on).