Does anyone trying to solve problems involving matrices with millions of entries not have some idea of the computational complexity of various operations they can do on them, given the nature of their matrices, etc.? I’d expect that either (a) someone has a small enough problem that the computational time is irrelevant and they should just arrive at the solution in whatever manner is conceptually clearest, given their current mathematical knowledge, or (b) the problem is going to take a lot of computation time, in which case it’s worth exploring to learn exactly what the fastest method will be.