the n-step transition matrix
The one-step matrix answers 'where might I be tomorrow?'. But you usually care about further horizons: where might I be in ten steps, or a thousand? The n-step transition matrix collects all those answers. Its entry P^(n)(i, j) is the probability of being in state j exactly n steps after starting in state i, no matter which path was taken in between.
The remarkable fact, given by the Chapman-Kolmogorov equations, is that you do not need any new theory to compute it: the n-step matrix is simply the ordinary matrix power, P^(n) = P^n. To find ten-step probabilities, raise P to the tenth power. Each matrix multiplication automatically sums over all the intermediate states for you. So a question that sounds combinatorial — count every n-step route and weight it by its probability — collapses into repeated matrix multiplication, something a computer does instantly.
This is the practical engine of the subject. Want the distribution after n steps from a known start? Read a row of P^n. Want it from an uncertain start with initial distribution pi_0? Compute the row vector pi_0 times P^n. And watching what happens to P^n as n grows large is exactly how you discover the chain's long-run equilibrium: for nice chains the rows of P^n all converge to the same stationary distribution, meaning the starting point is eventually forgotten.
Weather P has rows (0.8, 0.2) and (0.4, 0.6). Then P^2 has rows (0.72, 0.28) and (0.56, 0.44), and as n grows the rows of P^n both approach (2/3, 1/3). So after many days the chance of sun is about 2/3 regardless of today's weather — the n-step matrix reveals the long-run climate.
Raising P to higher powers and watching the rows converge reveals the chain's equilibrium.
P^(n) is the n-th matrix POWER, not P with each entry raised to the n-th power. Entrywise powering is a common beginner error and gives nonsense.