Exercise 12.12 A square matrix with non-negative entries is called doubly stochastic if all its row sums and column sums equal 1. If P is doubly stochastic, show that Pn is doubly stochastic for n ≥ 1 Let S = {1, 2, 3, 4, 5, 6} and P = 

1 2 1 2 0 0 0 0 0 0 1 0 0 0 1 3 0 0 1 3 1 3 0 0 0 0 1 2 1 2 0 0 0 0 0 0 1 0 0 0 0 1 0 
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.