In a real application, you don't even know the transition probabilities to begin with, so actually performing the computation is the least of your worries -- the first problem is coming up with numbers, which wasn't mentioned. :)
So I feel that if the goal is to give a conceptual understanding, linear algebra is overkill. If the goal is to show how it's done in practice then this isn't terribly useful since it doesn't even show the tip of the iceberg!