Question

Let W be the 5x5 matrix from the data above, where each entry is a probability between 0 and 1 rather than a percentage: 0.76 0.03 0.18 0.02 0.01 0.04 0.85 0.11 0.00 0.00 W 10.10 0.03 0.80 0.04 0.03 0.07 0.01 0.15 0.700.07 0.10 0.03 0.00 0.050.82 PROBLeM 2.1. Observe that Wn- n where n-1. Explain why this makes sense. 0.26 0.16 PROBLEM 2.2. Observe that Wrp ~ p where p-0.38|. Explain why this makes sense. 0.09 0.11 PROBLeM 2.3. Let I denote the 5x5 identity matrix (a) Find all solutions X to the equation W7-X. What is the rank of the matrix W - I? (b) Find all solutions ž to the equation W'x- X. What is the rank of the matrix WT - I? (c) Explain why it is not a coincidence that the rank of W - I and W I are the same. PRObLeM 2.4. Think of a process that Christopher might have used to calculate the per- centage of time that woozle spend doing each activity. If your answer involves solving a certain system of linear equations, explain how Christopher could anticipate that the system is consistent.

Answer 1

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