Unless you're unlucky, and $\lambda$ is an eigenvalue, this is just straightforward solving of a system of linear equations. There is lots of good software for doing this efficiently. Matlab can do it, and there is software in the "Numerical Recipes" book, or elsewhere. Don't try to write the code yourself, and don't try to find the inverse of the matrix.
If you're interested in finding eigenvalues, then there's lots of good software for that, too. Again, "Numerical Recipes" is a good place to start. Again, don't try to write the software yourself. Also, don't try to find the roots of the characteristic polynomial -- that's much more difficult than computing eigenvalues.